Fourier Series and the Fourier Transform
 4.4k Downloads
Abstract
Sections 10.1 and 10.2 are devoted to the general theory of orthogonal systems. Besides basic results (Bessel’s inequality, Riesz–Fischer theorem, etc.) we consider various examples of orthogonal systems (trigonometric system, Rademacher functions, Legendre polynomials, etc.). At the end of Sect. 10.2, we consider orthogonal series of independent functions, which play an important role in probability theory.
In Sects. 10.3 and 10.4 we discuss facts related to trigonometric Fourier series. For such series, we establish convergence conditions, the possibility of termwise integration, and the uniqueness theorem (for both summable functions and measures). We consider summation methods for Fourier series (including the classical methods of Fejér and Abel–Poisson) and the corresponding approximate identities. At the end of Sect. 10.4, we discuss comparatively recent results showing that the results concerning trigonometric series for functions of one variable cannot be carried over to Fourier series of functions of several variables. The rest of Sect. 10.4 is devoted to multiple Fourier series. Along with the counterparts of certain statements in the onedimensional case, we discuss some facts (failure of the localization principle, etc.) showing that certain classical results cannot be carried over to the multidimensional case.
Section 10.5 is devoted to the Fourier transform, which is one of the most important concepts in harmonic analysis. We consider both \(\mathcal{L}^{1}\) and \(\mathcal{L}^{2}\) theory of the Fourier transform, and, in particular, prove the inversion formula, Plancherel’s theorem, and the uncertainty principle. We also study the Fourier transforms of finite Borel measures. Using the Fourier transform, we prove that the \(\mathcal{L}^{1}\)norms of the Dirichlet kernels for balls (such kernels arise when studying the multiple Fourier series) have power rate of growth.
In Sect. 10.6, we discuss the Poisson summation formula and its applications. In particular, we show how this formula can be used to estimate the number of integer points in a ball (Gauss’s problem).
Keywords
Multiple Fourier Series Arbitrary Orthogonal System Trigonometric System Poisson Summation Formula Approximate Identity10.1 Orthogonal Systems in the Space Open image in new window
In the present section, we consider only the norm in the space Open image in new window . For brevity, we denote it by ∥⋅∥ without index.
10.1.1
From the continuity of the scalar product, it follows that the scalar multiplication of a series convergent in norm by a function can be carried out termwise, \(\langle\sum_{n=1}^{\infty} f_{n},g\rangle=\sum_{n=1}^{\infty}\langle f_{n},g\rangle\). To verify this, it is sufficient to pass to the limit in the equation \(\langle\sum_{n=1}^{k} f_{n},g\rangle=\sum_{n=1}^{k}\langle f_{n},g\rangle\) (the limit on the lefthand side of the equation exists since the series converges and the scalar product is continuous).
The reader can easily verify that if a measure is nondegenerate (more precisely, if there exist two disjoint sets of positive finite measure), then in each space Open image in new window with p≠2 the parallelogram identity is violated.
10.1.2
In the presence of a scalar product, as in a finitedimensional Euclidean space, we can introduce the notion of the angle between vectors. We are not going to do this in the general setting, instead restricting ourselves to the most important case where the angle is π/2. We introduce the following definition.
Definition
Functions Open image in new window are called orthogonal if 〈f,g〉=0.
Due to the scalar product, every ndimensional space L contained in Open image in new window is isomorphic (as a Euclidean space) to \(\mathbb{R}^{n}\) or \(\mathbb{C}^{n}\) (depending on the field of scalars under consideration). Therefore, we can speak of the orthogonal projection of a function f onto a subspace L. In particular, the projection of f onto the onedimensional subspace generated by the unit vector e, is 〈f,e〉e.
In the space Open image in new window , the families of pairwise orthogonal functions play a role similar to that of the orthogonal bases in finite dimensional Euclidean spaces.
Definition
A family of functions {e _{ α }}_{ α∈A } is called an orthogonal system (briefly, OS) if e _{ α }⊥e _{ α′} for α≠α′ and ∥e _{ α }∥≠0 for every α∈A. An orthogonal system is called orthonormal if ∥e _{ α }∥=1 for every α∈A.
It follows immediately from the Pythagorean theorem (1) that the functions from an OS are linearly independent. Obviously, dividing each element of an orthogonal system by its norm, we obtain an orthonormal system.
Let the functions e _{1},…,e _{ n } form an OS, and let L be the subspace generated by e _{1},…,e _{ n } (i.e., the set of all linear combinations of these functions). It is important to know how to find the best approximation to a given function f by elements of L. The following theorem gives a solution of this extremal problem.
Theorem
Thus, the function \(\sum_{k=1}^{n} c_{k}(f)e_{k}\) is the best approximation for f in the set L. The abovestated theorem can be regarded as a generalization of the following wellknown fact of school geometry: “the perpendicular dropped from a point f to L”, i.e., the difference \(f\sum_{k=1}^{n} c_{k}(f)e_{k}\), is shorter than any “slant” \(f\sum_{k=1}^{n}a_{k}e_{k}\).
Proof
10.1.3
The series with coefficients calculated by formula (2) play an important role, which justifies the following definition.
Definition
Let \(\{e_{n}\}_{n\in\mathbb{N}}\) be an orthogonal system, and let Open image in new window . The numbers c _{ n }(f) obtained by formula (2) are called the Fourier ^{3} coefficients, and the series \(\sum_{n=1}^{\infty} c_{n}(f)e_{n}\) is called the Fourier series of f with respect to the given OS.
As we will see, the Fourier series of an arbitrary function Open image in new window converges in the norm ∥⋅∥ (but not necessarily to f).
10.1.4
We do not yet know whether a Fourier series converges or, in the case of convergence, what its sum is. The following important theorem establishes that the sum of a Fourier series always exists. As a preliminary, we prove the following lemma.
Lemma
Proof
Theorem
(Riesz–Fischer^{4})
Proof
By Bessel’s inequality, we obtain \(\sum_{n=1}^{\infty}c_{n}(f)^{2}\e_{n}\^{2}\leqslant\f\^{2}<+\infty\), and so the series \(\sum_{n=1}^{\infty} c_{n}(f)e_{n}\) converges by the lemma. Let S be its sum. By the second assertion of the lemma, we have c _{ n }(f)≡c _{ n }(S). Therefore, the Fourier coefficients of the difference h=f−S are zero, i.e., h⊥e _{ n } for all n. □
10.1.5
Obviously, the sum of the Fourier series may not coincide with the function generating this series. For example, if we replace an OS e _{1},e _{2},… by the system e _{2},e _{3},… obtained by deleting the first vector, then the Fourier coefficients of the function e _{1} with respect to the new system are zeros, and e _{1} is not equal to the sum of its Fourier series (with respect to the new system).
Definition
An orthogonal system \(\{e_{n}\}_{n\in\mathbb{N}}\) is called a basis if every function in Open image in new window coincides with the sum of its Fourier series almost everywhere.
If \(\{e_{n}\}_{n\in\mathbb{N}}\) is a basis, then, by (1′), the relation \(f=\sum_{n=1}^{\infty} c_{n}(f)e_{n}\) implies that \(\f\^{2}=\sum_{n=1}^{\infty}c_{n}(f)^{2}\e_{n}\^{2}\). Thus, for a basis, the Bessel inequality becomes an equality. We will prove that this property characterizes a basis.
We introduce one more important property which, like Parseval’s identity, is characteristic for a basis.
Definition
Lemma
A family {f _{ α }}_{ α∈A } is complete if the set of all linear combinations of functions contained in this family is everywhere dense, i.e., if, for every function Open image in new window and every ε>0, there exists a linear combination \(g=\sum_{k=1}^{n} c_{k} f_{\alpha _{k}}\) such that ∥f−g∥<ε.
Proof
Theorem
(On the characterization of bases)
 (1)
the system \(\{e_{n}\}_{n\in\mathbb{N}}\) is a basis;
 (2)
for every function Open image in new window , Parseval’s identity \(\sum_{n=1}^{\infty}c_{n}(f)^{2}\e_{n}\^{2}=\f\^{2}\) holds;
 (3)
the system \(\{e_{n}\}_{n\in\mathbb{N}}\) is complete.
Proof
We prove the chain of implications (1)⇒(2)⇒(3)⇒(1).
(1)⇒(2) This implication was proved just after the definition of a basis.
(2)⇒(3) Assume that f⊥e _{ n }, i.e., c _{ n }(f)=0 for all n=1,2,…. By hypothesis, \(\f\^{2}=\sum_{n=1}^{\infty}c_{n}(f)^{2}\ e_{n}\^{2}=0\), which means that the system \(\{e_{n}\}_{n\in\mathbb{N}}\) is complete.
(3)⇒(1) Let Open image in new window . By the Riesz–Fischer theorem, f=g+h, where \(g=\sum_{n=1}^{\infty} c_{n}(f)e_{n}\) and h⊥e _{ n } for all n. Since the system is complete, we obtain that h=0 almost everywhere. Taking account of the arbitrariness of f, we obtain that the OS in question is a basis. □
Comparing the theorem with the preceding lemma, we see that the following statement is valid.
Corollary
An orthogonal system \(\{e_{n}\}_{n\in\mathbb{N}}\) is complete if and only if the set of all linear combinations of the functions contained in this system is everywhere dense.
10.1.6
We will see in the next section (see also Sect. 10.2) that it is often convenient to label naturally arising orthogonal systems not by positive integers but by some other indices. Therefore, it is useful to generalize the definition of the Fourier series and coefficients. Let {e _{ α }}_{ α∈A } be an arbitrary OS in the space Open image in new window , and let Open image in new window . As above, the numbers \(c_{\alpha }(f)=\frac{\langle f,e_{\alpha }\rangle}{\e_{\alpha }\^{2}}\) will be called the Fourier coefficients of the function f with respect to the given OS. Since Bessel’s inequality \(\sum_{k=1}^{n}c_{\alpha _{k}}(f)^{2}\e_{\alpha _{k}}\^{2}\leqslant\f\^{2}\) is valid for every finite set of indices α _{1},…,α _{ n }, the family {c _{ α }(f)^{2}∥e _{ α }∥^{2}}_{ α∈A } is summable (see Sect. 1.2.2). Therefore, the set A _{ f } of indices of the nonzero coefficients c _{ α }(f) is at most countable (see Sect. 1.2.2), which, after enumeration, can be written in the form {α _{1},α _{2},…}. By the Riesz–Fischer theorem, the series \(\sum_{k=1}^{\infty} c_{\alpha _{k}}(f)e_{\alpha _{k}}\) converges, and its sum will also be called the sum of the Fourier series of f with respect to {e _{ α }}_{ α∈A }. To verify that the sum is welldefined, we must prove that different enumerations of the set A _{ f } give the same sum. A change of enumeration of the set A _{ f } results in a series obtained by rearranging the terms of the series \(\sum_{k=1}^{\infty} c_{\alpha _{k}}(f)e_{\alpha _{k}}\). Therefore, it is sufficient to prove the following auxiliary statement.
Lemma
Proof
As in the case of sequences, a family {e _{ α }}_{ α∈A } is called a basis if every function is the sum of its Fourier series. It can easily be seen that the theorem on the characterization of bases and its corollary remain valid in the more general setting in question.
10.1.7
Theorem
If orthogonal systems \(\{e_{k}\}_{k\in\mathbb{N}}\) and \(\{g_{n}\}_{n\in\mathbb{N}}\) are complete, then the system \(\{h_{k,n}\}_{k,n\in\mathbb{N}}\) is also complete.
Proof
By induction, the statement just proved can obviously be carried over to the case of more than two orthogonal systems.
10.1.8
Lemma 10.1.4 shows that, for a given orthonormal system, an arbitrary sequence {a _{ n }}_{ n⩾1} satisfying the condition \(\sum_{n=1}^{\infty}a_{n}^{2}<+\infty\) can serve as the sequence of Fourier coefficients of a squaresummable function. It is natural to assume that the smaller the class of functions in question, the greater, in general, the rate of decrease of the Fourier coefficients. In Sect. 10.3, we will find more evidence for this conjecture. However, if, instead of squaresummable functions, we consider arbitrary bounded functions (assuming, naturally, that they belong to Open image in new window , i.e., that the measure μ is finite) then our conjecture is false: the Fourier coefficients of bounded functions tend to zero “no faster” than the Fourier coefficients of arbitrary functions from Open image in new window . A more precise formulation of this result of F.L. Nazarov^{6} [Na] is as follows.
Theorem
Let \(\{e_{n}\}_{n\in\mathbb{N}}\) be an orthonormal system in Open image in new window , μ(X)<+∞, such that ∫_{ X }e _{ n } dμ⩾β>0, where β does not depend on n. Then, for every series \(\sum_{n=1}^{\infty} a_{n}^{2}=1\) (a _{ n }>0), there exists a measurable function F _{ a } such that F _{ a }⩽1 and c _{ n }(F _{ a })⩾θ a _{ n } for all n (the coefficient θ>0 depends only on μ(X) and β).
We note that the condition ∫_{ X }e _{ n } dμ⩾β>0 is certainly fulfilled if the orthogonal system consists of uniformly bounded functions since 1=∫_{ X }e _{ n }^{2} dμ⩽∥e _{ n }∥_{∞}∫_{ X }e _{ n } dμ.
Proof
We consider only the real case, leaving the complex case to the reader (see Exercises 6 and 7).
EXERCISES
 1.

Supplement Lemma 10.1.4 by the following statement: if a system (in general, nonorthogonal) \(\{e_{n}\}_{n\in\mathbb{N}}\) is such that the inequality ∥a _{1} e _{1}+⋯+a _{ n } e _{ n }∥^{2}⩽a _{1}^{2}+⋯+a _{ n }^{2} is valid for all n and all scalars a _{1},…,a _{ n }, then the series \(\sum_{n=1}^{\infty} a_{n}e_{n}\) converges as soon as \(\sum_{n=1}^{\infty}a_{n}^{2}<+\infty\).
 2.

Let an orthonormal system \(\{e_{n}\}_{n\in\mathbb{N}}\) in Open image in new window be uniformly bounded. Prove that \(\int_{X} f\overline{e}_{n}\,d\mu\underset{n\to\infty}{\longrightarrow}0\) for every function f not only from Open image in new window but also from Open image in new window .
 3.

Let \(\{e_{n}\}_{n\in\mathbb{N}}\) be an orthonormal basis in Open image in new window , and let E⊂X be such that 0<μ(E)<+∞. Prove that \(\sum_{n=1}^{\infty}\int_{E}e_{n}^{2}\,d\mu\geqslant1\).
 4.

Supplement the previous exercise by proving that \(\sum_{n=1}^{\infty}e_{n}(x)^{2}=+\infty\) is valid almost everywhere if the σfinite measure μ is such that every set of positive measure can be partitioned into two sets of positive measure. Can this additional condition be dropped?
 5.

Let {φ _{ n }} be an orthonormal basis. Prove that the system of functions {ψ _{ n }} is complete if ∑_{ n }∥φ _{ n }−ψ _{ n }∥^{2}<1. If, in addition, we know that {ψ _{ n }} is an orthonormal system, then it is complete if ∑_{ n }∥φ _{ n }−ψ _{ n }∥^{2}<2. Hint. Assuming that a function f=∑c _{ n } φ _{ n } is orthogonal to all functions ψ _{ n }, estimate the norm of the difference f−∑_{ n } c _{ n } ψ _{ n } from above and from below.
 6.

Verify that Theorem 10.1.8 remains valid in the real case if the orthonormality condition is replaced by the condition from Exercise 1 (the quantities 〈F _{ a },e _{ n }〉 are estimated instead of the Fourier coefficients).
 7.

Generalize the result of the previous exercise to complex systems.
10.2 ^{⋆}Examples of Orthogonal Systems
Throughout this section, we consider the convergence of Fourier series only with respect to the Open image in new window norm, which is denoted by ∥⋅∥. Instead of Open image in new window , where \(X\subset\mathbb{R}^{m}\), we will write briefly Open image in new window , omitting the indication of a measure.
10.2.1
In the study of Fourier series, we may assume that the functions are defined on the intervals of the form (0,2ℓ), because the general case can be reduced to the case a=0 by a translation. It is often convenient to use a symmetric interval (−ℓ,ℓ).
In the following theorem, we establish one of the most important properties of the systems (T).
Theorem
The real and complex trigonometric systems form bases in Open image in new window .
Proof
The assertion of the theorem follows immediately from Corollary 10.1.5 the assumptions of which are fulfilled by Theorem 4 of Sect. 9.3.7. □
We will now give several examples that illustrate the importance of this formula.
Example 1
Example 2
As we have seen (see Corollary 9.2.4), 2πperiodic functions in Open image in new window , i.e., square integrable functions on (−π,π) are continuous in mean. By the closeness equation, we can obtain an exact value for the deviation of a function from its translation.
From this formula, the continuity in the mean, \(f_{h}\underset{h\to0}{\longrightarrow} f\), follows directly.
Example 3
We apply Parseval’s identity to prove an elegant inequality (see [EF]), which, in some cases, makes it possible to estimate from above the mean value of a function on an interval by its mean value on a smaller interval.
Example 4
The proof given by Hurwitz is analytic. It uses only the closeness equation and the formula for the area in terms of a curvilinear integral.
Let \(K\subset\mathbb{R}^{2}\) be a compact set whose boundary is a closed smooth curve. Without loss of generality, we may assume that the length of the curve is 2π. Let z(t)=(x(t),y(t)), 0⩽t⩽2π be the natural parametrization (see Sect. 8.2.3) of the curve ∂K. Then z(0)=z(2π) because the curve ∂K is closed and z′(t)≡1 because the parametrization is natural.
10.2.2
Example
10.2.3
The trigonometric system is closely connected with the orthogonal system \(\{z^{n}\}_{n\in\mathbb{Z}}\) in the space Open image in new window , where \(S^{1}=\{z\in\mathbb{C}\,\,z=1\}\) is the unit circle and σ is the arc length. Knowing that the trigonometric system is complete in Open image in new window , we use the change of variable z=e ^{ ix } (−π<x<π) and easily verify that the system \(\{z^{n}\}_{n\in\mathbb{Z}}\) is complete in Open image in new window . Therefore, every function f in this space is the sum of the series \(\sum_{n\in\mathbb{Z}} c_{n} z^{n}\), where \(c_{n}=\frac{1}{2\pi}\int_{S^{1}}f(z)\overline{z}^{n}\,d\sigma (z)\). The reader familiar with the theory of holomorphic functions will see that this formula coincides with the formula for the nth coefficient of the Laurent expansion of f in the annulus r<z<R, where r<1<R. Therefore, the Fourier series in the system \(\{z^{n}\}_{n\in\mathbb{Z}}\) can be regarded as the limit form of the Laurent series, when the annulus degenerates to a circle.
We consider an example connected with the system \(\{z^{n}\}_{n\in\mathbb{Z}}\). Let T:S ^{1}→S ^{1} be a rotation of the circle, i.e., the map z↦T(z)=ζz, where ζ∈S ^{1} is a fixed number. We now address the question of how much the points of the circle “mix” under the iterations of T. Does there exist an invariant subset of the circle, that is, a set which retains all of its points after rotation? More precisely, a set E⊂S ^{1} is called invariant if it differs from its image only on a set of measure zero, i.e., if χ _{ E }=χ _{ T(E)} almost everywhere. Of course, such sets exist: the circle S ^{1} and the set \(\{\zeta^{n}\}_{n\in\mathbb{Z}}\) are examples. It is easy to construct more examples of invariant sets of measure 2π or zero. Therefore, we are interested in the question of whether there are nontrivial invariant sets, i.e., sets satisfying the condition 0<σ(E)<2π. If ζ ^{ m }=1 for some m, then the map T is repeated after m iterations (T ^{ m+1}=T), and a nontrivial invariant subspace can easily be constructed. We leave this construction to the reader. However, if ζ is not a root of unity, then the map T has no nontrivial invariant sets (such maps are called ergodic). Let us prove this.
10.2.4
Theorem
The Legendre polynomials form a basis in the space Open image in new window .
Proof
As in the proof of Theorem 10.2.1, we use Corollary 10.1.5. We must verify that every function in Open image in new window can be approximated arbitrarily closely (in the Open image in new window norm) by linear combinations of polynomials P _{ n }, i.e., by arbitrary algebraic polynomials. This, however, has already been established in Corollary 9.2.3. □
Later on (see the corollary in Sect. 10.5.6) we prove that the system of functions h _{ n } is complete in Open image in new window or, equivalently, the system of polynomials H _{ n } is complete in Open image in new window .
10.2.5
In particular, the Rademacher functions form an orthonormal system in the space Open image in new window . Of course, this system is not complete: for example, the pairwise products r _{ j } r _{ k } are orthogonal to all Rademacher functions. To obtain a complete system containing the Rademacher functions, we proceed as follows. For every nonempty finite set \(A\subset\mathbb{N}\), we consider the function w _{ A }=∏_{ n∈A } r _{ n }. Furthermore, we will assume, by definition, that w _{∅}≡1. The functions w _{ A } are called the Walsh ^{9} functions. The Rademacher functions are the Walsh functions corresponding to the oneelement sets. By Eq. (1), the functions w _{ A } are pairwise orthogonal. The system of Walsh functions is complete in Open image in new window . To prove this, we need the following lemma.
Lemma
Let \(n\in\mathbb{N}\). The set of linear combinations of the functions w _{ A } such that A⊂{1,2,3,…,2^{ n }} coincides with the set of linear combinations of the characteristic functions of the intervals Δ_{ n,k }=(k2^{−n },(k+1)2^{−n }) for k=0,1,…,2^{ n }−1.
Proof
Let L _{1} and L _{2} be the linear spans of the first and second systems, respectively. Since the functions r _{1},…,r _{ n } are constant on the intervals Δ_{ n,k }, the Walsh functions in question are also constant on these intervals. Therefore, L _{1}⊂L _{2}. At the same time, the dimensions of L _{1} and L _{2} are, obviously, equal (to 2^{ n }). Hence it follows that L _{1}=L _{2}. □
Theorem
The system of Walsh functions is complete in the space Open image in new window .
Proof
We use Corollary to Theorem 10.1.5 on the characterization of bases. We will prove that every function f in Open image in new window can be approximated arbitrarily closely in norm by linear combinations of Walsh functions. If f is the characteristic function of an interval (p,q)⊂(0,1), then, for a given ε, we can find a large n such that p and q can be approximated by the points j/2^{ n } and k/2^{ n } within ε. Then ∥f−χ _{Δ}∥^{2}<2ε, where χ _{Δ} is the characteristic function of the interval (j/2^{ n },k/2^{ n }), which almost everywhere coincides with the sum \(\sum_{s=j}^{k1}\chi_{\Delta_{n,s}}\) equal, by the lemma, to a certain linear combination of Walsh functions. Being able to approximate the characteristic functions of the intervals, we can also approximate their linear combinations, i.e., the step functions. Now, we consider the general case. By Theorem 9.2.2, for each ε, we can find a step function g such that ∥f−g∥<ε. Approximating g within ε by a linear combination h of Walsh functions, we obtain ∥f−h∥⩽∥f−g∥+∥g−h∥<2ε. Since ε was arbitrary, this completes the proof. □
10.2.6
From the viewpoint of probability theory, the Rademacher functions give an example of a sequence of independent trials with two equiprobable outcomes (the simplest “Bernoulli scheme”). Here, a “simple” random event is a roll of a number x∈(0,1), and the probability that a point will fall in the interval (p,q) is the length of the interval. A “trial” consists of the calculation of the values of the Rademacher functions: the first trial is the calculation of r _{1}(x), the second trial is the calculation of r _{2}(x), etc. Taking into account the connection between the values of a Rademacher function at a given point and the binary digits of the point, we can replace r _{ n }(x) by ε _{ n }(x) (the binary digits of x).
Two centuries after Bernoulli, Borel proved a stronger statement.
Theorem
(Strong law of large numbers)
Proof
10.2.7
The theorem just proved admits various generalizations also called the strong laws of large numbers. The statements concerning sequences of independent functions with zero mean values (obviously, these functions form an orthogonal system) are of most interest. Before passing to this question, we consider an inequality playing a decisive role in the study of series of such functions.
Throughout this section, we consider real functions in the space Open image in new window , assuming that the measure μ is normalized (μ(X)=1).
Theorem
(Kolmogorov’s^{10} inequality)
Proof
We supplement the theorem (preserving the notation) and verify that, for a sequence of independent functions f _{ n } satisfying the assumptions of the theorem, the following statement is true.
Corollary
If \(A^{2}=\sum_{n=1}^{\infty}\int_{X}f_{n}^{2}\,d\mu<+\infty\), then the function \(S^{*}=\sup_{k\geqslant1}S_{k}=\sup_{k\geqslant1}S_{k}^{*}\) is summable and ∫_{ X } S ^{∗} dμ⩽2A.
Proof
10.2.8
The estimate for the integral of the function S ^{∗} established in the previous corollary leads to an important result concerning the behavior of the series \(\sum_{n=1}^{\infty} f_{n}\), which, in turn, implies a generalization of Borel’s theorem.
Theorem 1
Let \(\{f_{n}\}_{n=1}^{\infty}\) be a sequence of independent functions with zero means. If \(\sum_{n=1}^{\infty}\int_{X}f_{n}^{2}\,d\mu<+\infty\), then the series \(\sum_{n=1}^{\infty} f_{n}\) converges almost everywhere.
Proof
Corollary
Let \(\{f_{n}\}_{n=1}^{\infty}\) be a sequence of independent functions with zero means. If \(\sum_{n=1}^{\infty}\frac{1}{n^{2}}\int_{X} f_{n}^{2}\,d\mu<+\infty\), then \(\sigma _{n}=\frac{1}{n}\sum_{k=1}^{n}f_{k}\underset{n\to\infty}{\longrightarrow}0\) almost everywhere.
Proof
By the theorem, the sums \(T_{n}=\sum_{k=1}^{n}\frac{1}{k}f_{k}\) have a finite limit almost everywhere. The quantities \(\theta_{n}=\frac{1}{n+1}(T_{1}+\cdots+T_{n})\) have the same limit. Therefore, the difference T _{ n }−θ _{ n } tends to zero almost everywhere. At the same time, it is easy to verify that \(T_{n}\theta_{n}=\frac{1}{n+1}(f_{1}+\cdots+f_{n})\), which completes the proof. □
A similar statement can be obtained for an arbitrary orthogonal system if we drop the independence requirement and strengthen the restriction on the quantities ∥f _{ n }∥ (see Exercise 11).
If we impose quite natural additional restrictions on the independent functions f _{ n }, then the condition \(\sum_{n=1}^{\infty}\int_{X}f_{n}^{2}\,d\mu<+\infty\) will turn out to be not only sufficient but also necessary for the convergence of the series \(\sum_{n=1}^{\infty} f_{n}\) almost everywhere (or, equivalently by the zeroone law, on a set of positive measure).
Theorem 2
Let \(\{f_{k}\}_{k=1}^{\infty}\) be a sequence of independent bounded functions with zero means. If the series \(\sum_{k=1}^{\infty} f_{k}\) converges almost everywhere, then \(\sum_{k=1}^{\infty}\int_{X}f_{k}^{2}\,d\mu<+\infty\).
Proof
EXERCISES
 1.

Prove that the systems 1,cos x,cos 2x,… and sin x,sin 2x,… are complete in the space Open image in new window .
 2.

Let μ be a measure on the interval (−1,1) having density \(\frac{1}{\sqrt{1x^{2}}}\) with respect to Lebesgue measure. Prove that the functions T _{ n }(x)=cos(n arccos x) (n=0,1,2,…) form an orthogonal basis in the space Open image in new window . Verify that T _{ n } is an algebraic polynomial of degree n (a Chebyshev polynomial).
 3.

Prove that the functions x↦e ^{ x/2}(x ^{ n } e ^{−x })^{(n)} (n=0,1,…), called the Laguerre ^{11} functions, form an orthogonal system in the space Open image in new window .
 4.
 Let m be one of the digits 0,1,…,9, and let c _{ m }(x)=1 if the decimal expansion of the fractional part of x has the form 0.m… and c _{ m }(x)=0 otherwise. Verify that$$\frac{1}{n}\sum_{0\leqslant k<n}c_m \bigl(10^kx\bigr)\underset{n\to\infty}{\longrightarrow} \frac{1}{10}\quad\text{almost everywhere on}\ (0,1). $$
 5.

Generalize the result of the previous exercise by proving that almost all numbers x∈(0,1) are normal, i.e., for each \(p\in\mathbb{N}\), the pary expansion of x contains all digits (the numbers 0,1,…,p−1) “equally often”.
In Exercises 6–9, by r _{1},r _{2},…,r _{ n }… we denote the Rademacher functions.
 6.
 Use the Khintchine inequality to supplement Theorem 10.2.6 by proving thatfor p>1/2.$$\frac{r_1(x)+\cdots+r_n(x)}{n^p} \underset{n\to\infty}{\longrightarrow} 0\quad \text{almost everywhere on}\ (0,1) $$
 7.

Verify that the result of the previous exercise is false for p=1/2. Hint. Find the limits of the integrals \(\int_{0}^{1}e^{i\sigma _{n}(x)}dx\), where \(\sigma _{n}(x)=\frac{r_{1}(x)+\cdots+r_{n}(x)}{\sqrt{n}}\).
 8.

Show that if the sum of the series \(\sum_{n=1}^{\infty} a_{n}r_{n}\) is bounded almost everywhere on some nondegenerate interval, then \(\sum_{n=1}^{\infty}a_{n}<+\infty\).
 9.

Let \(f_{n}(x,y)=\sum_{j=1}^{n}r_{j}(x)r_{j}(y)\). Show that ∬_{ A×B } f _{ n }(x,y) dx dy⩽1 for any measurable sets A,B⊂(0,1), but nevertheless \(\iint_{(0,1)^{2}} f_{n}(x,y)\,dx\,dy\to+\infty\). Hint. Use Bessel’s inequality and the inequality from Exercise 7 of Sect. 9.1 with p=1.
 10.

Refine the assertion of Corollary 10.2.8 by proving that the functions σ _{ n } are dominated by a summable function.
 11.

Let \(\{f_{n}\}_{n=1}^{\infty}\) be an orthogonal system in Open image in new window such that \(\sum_{n=1}^{\infty}\frac{\f_{n}\^{2}}{n^{3/2}}<+\infty\). Prove that \(\frac{1}{n}(f_{1}+\cdots+f_{n})\,\underset{n\to\infty}{\longrightarrow}0\) almost everywhere on X and the function \(\sup_{n}\frac{1}{n}f_{1}+\cdots+f_{n}\) belongs to Open image in new window .
 12.

Verify that the assumptions of Theorem 2 of Sect. 10.2.8 can be weakened by replacing the convergence of the series \(\sum_{k=1}^{\infty} f_{k}\) almost everywhere by the boundedness of its partial sums at the points of a set of positive measure.
10.3 Trigonometric Fourier Series
The present and following sections are devoted to harmonic analysis. Without striving to expose this important and vast subject in its entirety, we restrict ourselves to the exposition of selected topics the choice of which is motivated only by the desire to demonstrate the methods developed above.
In Sect. 10.1, we established important properties of Fourier series in arbitrary orthogonal systems. Now, we consider the properties of Fourier series in trigonometric systems in more detail. This is historically the first example of an orthogonal system, and the problem of the representability of a function as the sum of a trigonometric series was one of the central problems in mathematics for nearly two hundred years.
Suffice to say that the lively discussion in the 18th century devoted to this problem provided an important impetus for the formulation of the modern concept of function. Riemann introduced his definition of an integral in connection with the study of trigonometric series, and Cantor, studying the uniqueness of the expansion of a function as a trigonometric series, came up with his foundation of set theory.
10.3.1
In conclusion, we touch on a question that may arise when solving the problem of the expansion of a function as a trigonometric series. Up to now, the choice of its coefficients have been dictated by geometric considerations presented in Sect. 10.1 and has led to formulas (2) and (2′). Can it happen that, for a different mode of convergence (e.g., pointwise or in measure) the coefficients of the trigonometric series must be chosen in a different way? It is easy to verify, however, that, under mild additional assumptions, there is essentially no freedom in the choice of the coefficients. Indeed, if, for example, a trigonometric series \(\sum_{k=\infty}^{\infty} c_{k} e^{ikx}\) converges to a function f almost everywhere or in measure and its partial sums S _{ n }(x)=∑_{k⩽n } c _{ k } e ^{ ikx } have a summable majorant, i.e., a function Open image in new window such that S _{ n }(x)⩽g(x) for all x∈(0,2π) and \(n\in\mathbb{N}\), then the coefficients of the series coincide with the Fourier coefficients of the function f, \(c_{k}\equiv\widehat{f}(k)\). Indeed, by Lebesgue’s theorem, the integral \(\widehat{f}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}f(x) e^{ikx}\,dx\) is the limit (as n→∞) of the integrals \(\frac{1}{2\pi}\int_{0}^{2\pi}S_{n}(x) e^{ikx}\,dx\), each of which is equal to c _{ k } for n⩾k.
10.3.2
Instead of functions defined only on the interval (0,2π), it will be more convenient for us to deal with 2πperiodic functions. Since every function defined on (0,2π) can be extended to a periodic function, we will assume in what follows that all functions in question are periodic (in the sequel, periodicity means 2πperiodicity). Being summable on an interval of length 2π, such functions are summable on each finite interval. We will repeatedly use the fact that the integral \(\int_{a}^{a+2\pi}f(x)\,dx\) does not depend on the parameter a (the reader is invited to prove this independently). Often, especially when dealing with odd and even functions, it is more convenient to integrate over the interval (−π,π) in formulas (2) and (2′). By \(\widetilde{C}\) and \(\widetilde{C}^{r}\) (1⩽r⩽+∞), we denote the classes of periodic functions that are continuous and, respectively, r times continuously differentiable on \(\mathbb{R}\); by Open image in new window , we denote the class of periodic functions summable on (−π,π) with power p⩾1. For a function Open image in new window , by ∥f∥_{ p } we mean the Open image in new window norm of its restriction to (−π,π).
 (a)
\(\widehat{f}(n)\leqslant\frac{1}{2\pi}\f\_{1}\) (see formula (2′)).
 (b)
\(\widehat{f}(n)\underset{n\to+\infty}{\longrightarrow}0\) (see the Riemann–Lebesgue theorem).
 (c)

\(\widehat{f_{h}}(n)=e^{inh}\widehat{f}(n)\).
 (d)
 If a periodic function f is absolutely continuous on \(\mathbb{R}\) (in particular, if it is piecewise differentiable), then(for the proof, it is sufficient to integrate by parts). In particular, \(\widehat{f}(n)=o(1/n)\). We note a weak version of this estimate for a function of bounded variation.$$\widehat{f'}(n)=in\widehat{f}(n)\quad(n\in\mathbb{Z}) $$
 (d′)
 If f is a function of bounded variation on the interval [0,2π], then \(\widehat{f}(n)=O(1/n)\). Indeed, integrating by parts (see Sect. 4.11.4), we obtain$$\begin{aligned} 2\pi\widehat{f}(n)=&\int_0^{2\pi}f(x) e^{inx}\,dx= f(x)\frac{e^{inx}}{in}\bigg\vert _0^{2\pi}+ \frac{1}{in}\int_0^{2\pi}e^{inx} \,df(x)\\ =&O \biggl(\frac{1}{n} \biggr). \end{aligned}$$
 (e)
 Let Open image in new window . Then(for the definition of the convolution of periodic functions, see Sect. 7.5.5). The proof is obtained by direct calculation using the change of the order of integration,$$\widehat{f*g}(n)=2\pi\widehat{f}(n)\cdot\widehat{g}(n)\quad\text{for all}\ n \in\mathbb{Z} $$$$\begin{aligned} \widehat{f*g}(n)=&\frac{1}{2\pi}\int_{\pi}^{\pi}(f*g) (x) e^{inx}\,dx\\=& \frac{1}{2\pi}\int_{\pi}^{\pi} \biggl(\int_{\pi}^{\pi} f(xt)g(t)\, dt \biggr)e^{inx}\,dx \\=&\frac{1}{2\pi}\int_{\pi}^{\pi} g(t) e^{int} \biggl(\int_{\pi}^{\pi} f(xt) e^{in(xt)}dx \biggr)dt \\=&\frac{1}{2\pi}\int_{\pi}^{\pi} g(t) e^{int} \biggl(\int_{\pi}^{\pi} f(u) e^{inu}du \biggr)dt=2\pi\widehat {g}(n)\cdot\widehat{f} (n). \end{aligned}$$
10.3.3
The problem of the Fourier series expansion of a function is rather complicated and has a long history. The famous work “The analytical theory of heat” by Fourier, in which the series that were later named after him were first studied and used systematically, did not contain an explicit formulation of a condition providing the expandability of a function as a Fourier series. Such criteria arose later. Still later it became clear that the Fourier series of a continuous function can diverge at some points, and, as Kolmogorov proved, the Fourier series of a summable function can diverge everywhere.
So far, even knowing that a Fourier series of a differentiable function converges at a point, we cannot be sure that its sum coincides with the value of the function.
At the moment, we know (see Sect. 10.2.1) that if f is a squaresummable function, then series (1′) converges in the Open image in new window norm and its sum is equal to f. If a function f is only assumed to be summable, the question of the convergence of a Fourier series (pointwise, in an Open image in new window norm, or in some other sense) remains open for the time being.
Thus, the general theorems connected with the use of approximate identities cannot be applied here. This is the cause of considerable difficulties in the study of the convergence of Fourier series. Here, we meet not just technical questions, but those of a fundamental nature. We will see later that the proofs of Theorems 2 and 3 of Sect. 9.3.7 cannot be carried over to convolutions with Dirichlet kernels.
The partial sums of the Fourier series are calculated by formula (5), and so depend on the values of the function on an interval of length 2π. It is all the more surprising that, as we will now verify, the convergence of the Fourier series at a point x and the value of its sum are local properties of the function, i.e., they are preserved under an arbitrary change of the function outside an arbitrarily small neighborhood of the point. More formally, we have the following.
Theorem
(Riemann’s localization principle)
If functions Open image in new window coincide in a neighborhood of a point x, then their Fourier series have the same behavior at x, S _{ n }(f _{1},x)−S _{ n }(f _{2},x)→0 as n→∞.
Proof
10.3.4
Among a great variety of convergence tests for Fourier series, we mention only two of the most applicable ones, the Dini^{12} test and the Dirichlet–Jordan^{13} test. They supplement each other and can be applied to a wide range of cases.
First, we establish a useful property of the Dirichlet kernel.
Lemma
Proof
Theorem
(Dini test)
Proof
Theorem
(Dirichlet–Jordan test)
If a periodic function f has bounded variation on the interval [−π,π], then, for each \(x\in\mathbb{R}\), the Fourier series of f converges to the average (f(x+0)+f(x−0))/2. Moreover, \(S_{n}(f,x)\leqslant\sup_{\mathbb{R}}f+2\mathbf{V}_{\pi}^{\pi}(f)\).
We remark that the convergence of a Fourier series at a point x is preserved by the localization principle if we assume that f has bounded variation only locally, in a neighborhood of this point.
Proof
If we sum them for k=0,1,…,n and for k=−1,…,−n separately, it becomes clear that the Dini condition implies the convergence of not only the symmetric sums \(\sum_{k=n}^{n}\widehat{f}(k) e^{ikx}\), but also the “onesided” sums \(\sum_{k=0}^{n}\widehat{f}(k) e^{ikx}\) and \(\sum_{k=n}^{1}\widehat{f}(k) e^{ikx}\). In other words, the Dini condition ensures the convergence of each of the series \(\sum_{k=0}^{\infty}\widehat{f}(k) e^{ikx}\) and \(\sum_{k=\infty }^{1}\widehat{f}(k) e^{ikx}\). In particular, it ensures the convergence of the series \(\sum_{n\in\mathbb{Z}}\text{sign}(n)\widehat{f}(n) e^{inx}\) called the conjugate to series (1′).
10.3.5
We give some examples of Fourier series expansions.
Example 1
Considering the Fourier series expansion of the function equal to x ^{2} on [−π,π] at the point π, we again obtain the Euler identity \(\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}\) (see Example 1 of Sect. 10.2.1 or Example 2 of Sect. 4.6.2).
Example 2
Example 3
We remark that the series just constructed does not converge absolutely. It can be proved (see [PS], Part 1, Problem 129) that it is impossible to construct an absolutely convergent series with the property in question.
10.3.6
As we have already noted, the Fourier series of a summable, or even of a continuous, function may diverge (see also Sect. 10.3.9). However, such a series has the remarkable property that it can be integrated termwise over an arbitrary finite interval without worrying about convergence.
Theorem 1
Proof
Theorem 1 allows us to considerably strengthen the assertion on the completeness of the trigonometric system, according to which two functions of the class Open image in new window that have the same Fourier coefficients coincide almost everywhere. Now, we can extend this result to the class Open image in new window .
Corollary 1
Functions Open image in new window having the same Fourier coefficients coincide almost everywhere on \(\mathbb{R}\).
Proof
By the theorem, the integrals of f and g are equal on every finite interval. Therefore, (see Corollary 4.5.4) f and g coincide almost everywhere. □
Corollary 2
For every function Open image in new window , the series \(\sum_{n=1}^{\infty} b_{n}(f)/n\) converges.
We recall that \(b_{n}(f)=\frac{1}{\pi}\int_{\pi}^{\pi} f(x)\sin nx\,dx= i(\widehat{f}(n)\widehat{f}(n))\) is the Fourier sine coefficient of f.
Proof
Corollary 2 gives a necessary condition for a trigonometric series \(\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)\) to be a Fourier series. The everywhere convergent series \(\sum_{n=2}^{\infty}(\sin nx)/\ln n\) does not satisfy this condition and, therefore, cannot be the Fourier series of a summable function. It is interesting to note that, in contrast to the sine coefficients, the cosine coefficients can tend to zero arbitrarily slowly. For example, the series \(\sum_{n=2}^{\infty}(\cos nx)/\ln n\) is the Fourier series of a summable function (see Theorem 10.4.2).
The relation obtained in Theorem 1 can be regarded as a new version of Parseval’s identity in which the assumption about one function is weakened (it belongs to Open image in new window but not to Open image in new window ) and the assumption concerning the other is strengthened considerably (it is the characteristic function of an interval). At the same time, the proof of the theorem uses the properties of the function χ only partially. This makes it possible to extend considerably the applicability conditions of Parseval’s identity.
Theorem 2
The class of functions with uniformly bounded partial sums of Fourier series is sufficiently wide. In particular, it contains all smooth functions on [−π,π]. As follows from the Dirichlet–Jordan test, this class also contains all functions with finite variation on [−π,π] (see also Exercises 9 and 10).
The assumption that the function g is bounded is superfluous (see Exercise 8 or Fejér’s theorem in Sect. 10.4).
Proof
10.3.7
To obtain a further generalization of the uniqueness theorem for Fourier series, (see Corollary 1 of the previous section), we introduce the notion of Fourier coefficients and Fourier series for a measure.
Definition
Theorem
Let μ and ν be finite Borel measures on the interval [−π,π] satisfying the condition μ({π})=ν({π})=0. If the Fourier coefficients of these measures coincide, then the measures also coincide.
Proof
Generalizations of this theorem are given in Sects. 10.4.7, 11.1.9, and 12.3.3.
10.3.8
Considering Fourier series with coefficients that tend to zero sufficiently fast, we must take into account that if the Fourier series of a function f converges uniformly, then its sum coincides with f almost everywhere by the uniqueness theorem. Therefore, if the Fourier series of a continuous function converges uniformly, then its sum coincides with the function. Taking this into account, we consider only continuous functions in the theorems of this section. Lifting the assumption of continuity, we must replace the equality of a function and its Fourier series by their equivalence.
The repeated application of the relation \(\widehat{f'}(n)=in\widehat {f}(n)\) (see property (d) of Sect. 10.3.2) shows that the Fourier coefficients of a function f of class \(\widetilde{C}^{r}\) satisfy the relation \(\widehat{f}(n)=o(n^{r})\) as n→+∞. The converse is “almost true”: if \(\widehat{f}(n)=O(n^{r2})\) for some \(r\in\mathbb{N}\), then the continuous function f coincides with a function of class \(\widetilde{C}^{r}\). Indeed, the series \(\sum_{n}\widehat{f}(n) e^{inx}\) converges uniformly, and, by the above remark, its sum coincides with f. Moreover, since the coefficients decrease fast, the Fourier series admits rfold differentiation, which implies that \(f\in\widetilde{C}^{r}\). For infinitely smooth functions, this gives a complete description.
Theorem 1
In order that a function \(f\in\widetilde{C}\) be infinitely differentiable it is necessary and sufficient that the limit relation \(n^{r}\widehat{f}(n)\to0\) as n→+∞ be fulfilled for every \(r\in\mathbb{N}\).
The smaller class of holomorphic periodic functions can also be well described in terms of the Fourier coefficients: these coefficients must tend to zero not slower that a geometric sequence. We note that a periodic function f is analytic at all points of the line \(\mathbb{R}\) if and only if, on \(\mathbb{R}\), the function f coincides with a function holomorphic in some horizontal strip \(\{z\in\mathbb{C}\,\,\mathcal{I} m\,z<L\}\). In the proof of the following theorem, we use some elementary properties of holomorphic functions (see, for example, [Ca]).
Theorem 2
 (a)
there is a function F holomorphic in a strip \(\mathcal{I} m\,z<L\) and coinciding with f on the real axis;
 (b)
the relation \(\widehat{f}(n) = O (e^{an} )\) as n→+∞ holds for every a∈(0,L).
Proof
The coefficients with negative indices can be estimated in the same way, only in this case the rectangle is replaced by a rectangle symmetric with respect to the real axis.
(b)⇒(a). The series \(\sum_{n=\infty}^{\infty}\widehat{f}(n) e^{inz}\) converges uniformly in the strip \(\mathcal{I} m\,z\leqslant a\) if 0<a<L. By Weierstrass’s theorem the sum of the series is holomorphic in the strip \(\mathcal{I} m\,z<L\) and coincides with the function f on the real axis. □
10.3.9
In the above example, we could select a subsequence \(\{S_{n_{k}}(f,0)\}\) that tends to +∞. As we will see in Sect. 10.4.1, it is impossible to construct a continuous function for which \(S_{n}(f,0)\underset{n\to\infty}{\longrightarrow}+\infty\). We also remark that, in the above example, estimate (6) for the Fourier sums is almost attained (in order) on the sequence {n _{ k }} (see also Exercise 13).
By a slightly more complicated construction it is possible to give an example of a continuous function whose Fourier series diverges on a countable set. Must this series converge almost everywhere? This famous problem was open for more than half a century. It was answered in the affirmative only in 1966 by L. Carleson.^{14} It turned out that the Fourier series of an arbitrary function of class Open image in new window (for example a continuous function) converges to the function almost everywhere (see [C]). Since that time, several modifications and strengthenings of the original proof have been obtained, but all of them are quite difficult and lie far beyond the scope of this book.
10.3.10
Theorem
(Denjoy^{15}–Luzin)
In particular, if a trigonometric series converges absolutely on a set of positive measure, then it converges uniformly on \(\mathbb{R}\), and, therefore, is the Fourier series of its sum.
Proof
EXERCISES
 1.

Let Open image in new window . By the method used in the second proof of the Lebesgue–Riemann theorem, prove that \(\widehat{f}(n)\leqslant\frac{1}{2}\ff_{\tau}\_{1}\), where f _{ τ } is the translation of the function f by the vector τ=πn/∥n∥^{2}.
 2.

Prove that the Fourier sums of the function \(f(x)=\sum_{k=1}^{\infty} 2^{k}\cos kx\) provide almost the best uniform approximations for f, more precisely, ∥f−S _{ n }(f)∥_{∞}⩽3∥f−T∥_{∞} for every trigonometric polynomial T of order n.
 3.

Show by example that an absolutely continuous function may not satisfy Dini’s condition.
 4.

Verify by examples that neither of the Dini and Dirichlet tests implies the other.
 5.
 Prove (see [HR], Sect. 6.7) thatif the function f in Open image in new window is continuous at x. Hint. Verify that the functions \(\frac{1}{2} (D_{n} (x+\frac{\pi}{2n} )+D_{n} (x\frac{\pi}{2n} ) )\) form a periodic approximate identity.$$\frac{1}{2} \biggl(S_n \biggl(f,x+\frac{\pi}{2n} \biggr)+ S_n \biggl(f,x\frac{\pi}{2n} \biggr) \biggr) \underset{n\to\infty}{ \longrightarrow} f(x) $$
 6.

Verify that Open image in new window belongs to the class Open image in new window if and only if the series \(\sum_{n=\infty}^{\infty}\widehat{f}(n)^{2}\) converges.
 7.

Prove that a function \(f\in\widetilde{C}\) is the restriction of an entire function to \(\mathbb{R}\) if and only if \(\root{n}\of{\widehat{f}(\pm n)}\to0\) as n→∞.
 8.

Prove that if the sequence of partial sums of a Fourier series is uniformly bounded, then the function belongs to Open image in new window .
 9.
 Let Open image in new window . Prove that the Fourier sums S _{ n }(hg,x) of the product hg are uniformly bounded (with respect to n and x) if the function g has the same property and the function h satisfies the Dini condition uniformly:$$\int_{\pi}^{\pi} \bigg\frac{h(u)h(x)}{ux} \bigg\,du \leqslant \text{const}\quad\text{for all}\ x. $$
 10.

Assume that a function f (possibly discontinuous) is such that the interval [−π,π] can be divided into a finite number of intervals inside each of which the function f satisfies the Lipschitz condition of order α>0. Prove that the Fourier sums S _{ n }(f,x) are uniformly bounded with respect to x and n.
 11.

Prove that \(\int_{\pi}^{\pi}D_{n}(u)\,du=\frac{4}{\pi^{2}}\ln n+O(1)\).
 12.

Supplement inequality (6) by proving that ∥f−S _{ n }(f)∥_{∞}=o(lnn) as n→∞ if \(f\in\widetilde{C}\). Verify that, for bounded functions, this refinement is, generally, false. Hint. Modify the Schwartz example by putting f(t)=sinn _{ k } t on the interval [t _{ k },t _{ k−1}].
 13.

Modifying the Schwartz example, verify that the result of the previous exercise is precise, i.e., that for every sequence ε _{ n }↓0, there is a function \(f\in \widetilde{C}\) such that S _{ n }(f,0)⩾ε _{ n }lnn along some sequence of indices n _{ k }→∞.
 14.

Show that the convergence of a Fourier series of a function \(f\in \widetilde{C}\) does not imply the convergence of the Fourier series of the function f ^{2}. Hint. Modifying the Schwartz example, construct a nonnegative even function \(F\in\widetilde{C}\), F(0)=F(±π)=0, with Fourier series divergent at zero and consider an odd function f equal to \(\sqrt{F}\) on [0,π].
 15.

Prove that \(a_{n},b_{n}\underset{n\to\infty}{\longrightarrow}0\) if the sums a _{ n }cosnx+b _{ n }sinnx converge to zero on a set of positive measure.
 16.

Find a set E⊂(0,2π) of the cardinality of the continuum and a sequence n _{ k }→+∞ such that sinn _{ k } x⇉0 on the set E.
 17.
 Consider the series \(\sum_{n=1}^{\infty}\sin(n!\pi x)\).
 (a)
Prove that the series converges at the points x=sin1, x=cos1, \(x=\frac{2}{e}\), and their multiples and converges at the points ke (\(k\in\mathbb{N}\)) only for odd k. Is the convergence absolute?
 (b)
Prove that the given series diverges at the points x=sinh 1 and \(x=\frac{1}{2}\text{cosh}\,1\).
 (c)
Find a set of the cardinality of the continuum at all points of which the given series converges.
 (a)
 18.

Prove that \(\frac{1}{n}S'_{n}(f,x)\underset{n\to\infty}{\longrightarrow} \frac{1}{\pi}(f(x+0)f(x0))\) if the periodic function f has a bounded variation on the interval [−π,π].
10.4 ^{⋆}Trigonometric Fourier Series (Continued)
10.4.1
The fact that a Fourier series may diverge, even at points of continuity, suggests that we might obtain information on its behavior if we consider a weaker definition of convergence than the classical one. One of the possible approaches is to investigate the convergence of the arithmetic means of the partial sums rather than the partial sums themselves. The limit of a sequence {a _{ n }} in the sense of arithmetic means or in the sense of Cesaro^{16} is, by definition, the limit \(\lim_{n\to\infty}\frac{1}{n}(a_{0}+\cdots+a_{n1})\). It can exist even in the case where the sequence itself diverges, for example, if a _{ n }=(−1)^{ n }. At the same time, if \(a_{n}\underset{n\to\infty}{\longrightarrow} a\), then also \(\frac{1}{n}(a_{0}+\cdots+a_{n1})\underset{n\to\infty}{\longrightarrow} a\) (the permanence of the method of arithmetic means). For numerical series, this approach leads to the concept of a generalized sum of a series. We say that a series Cesaro converges to a number C if the limit of the partial sums of the series in the sense of Cesaro is equal to C.
Theorem
(Fejér)
 (a)
if the limits L _{±}=lim_{ t→x±0} f(t) exist and are finite, then \(\sigma _{n}(f,x)\underset{n\to\infty}{\longrightarrow}\frac{L_{+}+L_{}}{2}\);
 (b)
if \(f\in\widetilde{C}\), then \(\sigma _{n}(f)\underset{n\to\infty}{\rightrightarrows} f\) on \(\mathbb{R}\);
 (c)
if Open image in new window for some p∈[1,+∞), then \(\\sigma _{n}(f)f\_{p}\underset{n\to\infty}{\longrightarrow}0\);
 (d)
\(\sigma _{n}(f)\underset{n\to\infty}{\longrightarrow} f\) almost everywhere.
Proof
In the case where the limit lim_{ t→x } f(t) exists and is finite, both assertions (a) and (b) are special cases of Theorem 7.6.5 (in view of the remarks to it) for m=1, \(T=\mathbb{N}\) and t _{0}=+∞. If the onesided limits are distinct, then we must use the fact that the Fejér kernels are even and apply the result obtained to the function f _{0}(u)=(f(x+u)+f(x−u))/2, which tends to (L _{+}+L _{−})/2 as u→0.
Assertion (c) is already known. It is a special case of Theorem 2 of Sect. 9.3.7.
By statement (a) of the theorem and the permanence of the method of arithmetic means, we are now able to answer the question we posed before taking up the investigation of the convergence of Fourier series (see Sect. 10.3.3): if the Fourier series of a summable function f converges at a point of continuity of f, then the sum of the Fourier series is necessarily equal to the value of f at this point.
Since the convolution f∗Φ_{ n } is a trigonometric polynomial, the Fejér theorem supplements the Weierstrass theorem (see Corollary 7.6.5) by providing specific approximating polynomials.
Remark
As we have verified, the Fejér sums σ _{ n }(f) have the obvious advantage over the partial sums S _{ n }(f) of the Fourier series that they approximate an arbitrary summable function in the integral metric and converge uniformly to f if f is continuous. It should be mentioned, however, that there is a price to pay for the universality of the Fejér sums: they cannot converge to the function rapidly (see Exercise 2). Therefore, if a Fourier series converges rapidly, then the Fejér sums are a poorer approximation of the function than the Fourier sums (see Exercise 2, Sect. 10.3).
In some cases, Fejér sums allow one to obtain an additional information concerning the behavior of Fourier sums.
Corollary 1
The Fourier series of an absolutely continuous periodic function f converges to f uniformly on \(\mathbb{R}\).
Proof
Corollary 2
The coefficient C _{ α } depends only on α (for the case α=1, see Exercise 3).
Proof
10.4.2
Before passing to the study of the sum of series (4), we prove a lemma on numerical series.
Lemma
 (1)
c _{ n−1}⩾c _{ n }⩾0 for all \(n\in\mathbb{N}\);
 (2)
\(\sum_{k=1}^{\infty} k(c_{k1}2c_{k}+c_{k+1})=c_{0}\).
Proof
We put b _{ n }=c _{ n−1}−c _{ n }. The convexity implies that b _{ n }⩾b _{ n+1}, and, consequently, b _{ n }⩾0 (because \(b_{n}\underset{n\to\infty}{\longrightarrow}0\)), which proves the first statement.
Theorem
Let the coefficients of series (4) form a convex sequence tending to zero. Then the sum f of the series is nonnegative and summable on (−π,π) and series (4) is the Fourier series of f.
Proof
A sequence of coefficients satisfying the conditions of the theorem can tend to zero arbitrarily slow (see Exercise 4). For example, since the sequence {1/lnn}_{ n⩾2} is convex, the theorem we have just proved implies that the sum of the series \(\sum_{n=2}^{\infty}\frac{\cos nx}{\ln n}\) belongs to Open image in new window and the series itself is the Fourier series of its sum. In this connection, we recall that the everywhere convergent series \(\sum_{n=2}^{\infty}\frac{\sin nx}{\ln n}\) is not a Fourier series, as established in Sect. 10.3.6.
10.4.3
The Fejér method is based on Cesaro’s generalization of the sum of a numerical series. Other generalizations of the concept of the sum of a series are also possible. One of them is based on the following wellknown Abel theorem for numerical series: if a series \(\sum_{n=1}^{\infty} c_{n}\) converges to the sum S, then the sum of the series \(\sum_{n=1}^{\infty} e^{\varepsilon n}c_{n}\) tends to S as ε→+0. This limit can exist also for a divergent series, and, therefore, can be regarded as a generalized sum of the series in question.
Let M be a decreasing function summable on [0,+∞), and let M(0)=lim_{ u→0} M(u)=1. It is clear that M⩾0 and the series \(\sum_{n\in\mathbb{Z}}M(\varepsilon n)\) converges for every ε>0.
To study the behavior of the sums S _{ M,ε }(f) as ε→+0, we impose an additional constraint on the function M. We will assume that it is convex on [0,+∞). Then, the sequence {M(εn)}_{ n⩾0} is convex and ω _{ ε }(x)⩾0 by Theorem 10.4.2.
Lemma
Let M be a continuous function convex on [0,+∞), tending to zero at infinity, and let M(0)=1. Then the functions ω _{ ε } form a periodic approximate identity with the strong localization property as ε→+0. Moreover, there exist even functions Ω_{ ε } that decrease on [0,π], majorize ω _{ ε }, and satisfy the inequality \(\int_{\pi}^{\pi}\Omega_{\varepsilon }(x)\,dx\leqslant2\).
For the definition of the strong localization property, see Sect. 7.6.5.
Proof
Now, we are able to study the behavior of the sums S _{ M,ε }(f,x).
Theorem
 (a)
if the limits L _{±}=lim_{ t→x±0} f(t) exist and are finite, then \(S_{M,\varepsilon }(f,x)\underset{\varepsilon \to0}{\longrightarrow}\frac{L_{+}+L_{}}{2}\);
 (b)
if \(f\in\widetilde{C}\), then \(S_{M,\varepsilon }(f)\underset{\varepsilon \to0}{\rightrightarrows} f\) on \(\mathbb{R}\);
 (c)
if Open image in new window for some p∈[1,+∞), then \(\S_{M,\varepsilon }(f)f\_{p}\underset{\varepsilon \to0}{\longrightarrow}0\);
 (d)
\(S_{M,\varepsilon }(f,x)\underset{\varepsilon \to0}{\longrightarrow} f(x)\) almost everywhere.
Proof
The proof can be obtained by repeating verbatim the proof of the Fejér theorem. Indeed, the proof of the Fejér theorem uses only the fact that the sum σ _{ n }(f) is the convolution of the function f and an even approximate identity satisfying the strong localization property and having a humpshaped majorant, which, by the lemma, is also valid for the sum S _{ M,ε }(f). □
10.4.4
The rest of the section is devoted to multiple trigonometric Fourier series, i.e., to series in the system \(\{e^{i\langle n,x\rangle}\}_{n\in\mathbb{Z}^{m}}\). As was mentioned in Sect. 10.2.2, this system is an orthogonal basis in the space Open image in new window . Therefore, the Open image in new window theory of multiple Fourier series is a special case of the general theory where the convergence of these series is in the Open image in new window norm, and we will not touch on this here. The situation is completely different in regard to other forms of convergence. The problems arising here are connected with the fact that there is no preferred definition of the sum of a multiple series. It is possible to find the limit of the partial sums over unboundedly expanding balls, cubes, parallelepipeds, etc. It turns out that the answers to most problems depend essentially on the choice of the definition of a partial sum. We will not dwell on this topic, instead confining ourselves to partial sums over rectangles. Even in this case, for m>1, there are phenomena that did not occur in the onedimensional situation.
When solving the problem of expanding a periodic function in the Fourier series \(\sum_{n\in\mathbb{Z}^{m}}c_{n} e^{i\langle n,x \rangle}\), as in the onedimensional case, there is no freedom in the choice of coefficients. The reasoning used at the end of Sect. 10.3.1 also remains also in the case in question. More precisely, let \(S_{j}(x) =\sum_{n\in A_{j}}c_{n}e^{i\langle n,x\rangle}\) be the partial sums of this series corresponding to expanding bounded sets \(A_{j}\subset \mathbb{R}^{m}\) such that \(\bigcup_{j=1}^{\infty} A_{j}=\mathbb{R}^{m}\). If \(S_{j}\underset{j\to\infty}{\longrightarrow} S\) almost everywhere or in measure, then the function S can be called the sum of the series. If, in addition, the partial sums S _{ j } are dominated by some function g in Open image in new window , then the coefficients of the given trigonometric series are determined uniquely. Indeed, as follows from Lebesgue’s theorem, \(\widehat{S}(n)=\lim_{j\to\infty}\widehat{S}_{j}(n)=c_{n}\). Therefore, under the present hypothesis, the expansion of a function in a multiple Fourier series is unique.
For absolutely convergent Fourier series, all definitions of partial sums give the same result because, in this case, the terms of the Fourier series form a summable family. Moreover, an absolutely convergent trigonometric series is, obviously, the Fourier series of its sum. Since the trigonometric system is complete (see Theorem 10.2.2) the sum of an absolutely convergent Fourier series coincides with the function almost everywhere, and if the function is continuous, it converges everywhere (in the onedimensional case, this has been noted at the beginning of Sect. 10.3.8). As we will soon verify, the Fourier series of smooth functions converge absolutely.
 (a)
\(\widehat{f}(n)\leqslant\frac{1}{(2\pi)^{m}}\f\_{1}\);
 (b)
\(\widehat{f}(n)\to0\) as ∥n∥→+∞;
 (c)
the Fourier coefficients of a translation f _{ h } (\(h\in\mathbb {R}^{m}\)), i.e., of the function f _{ h }(x)=f(x−h), are connected with the Fourier coefficients of f by the formulas \(\widehat{f}_{h}(n)=e^{i\langle n,h\rangle}\widehat{f}(n)\);
 (d)
if \(f\in\widetilde{C}^{r}(\mathbb{R}^{m})\) and \(g=\frac{\partial^{r} f}{\partial x_{j_{1}}\ldots\partial x_{j_{r}}}\), then \(\widehat{g}(n)=i^{r}n_{j_{1}}\cdots n_{j_{r}}\,\widehat{g}(n)\);
 (e)
\(\widehat{f*g}(n)=(2\pi)^{m} \widehat{f}(n)\cdot\widehat{g}(n)\) for all functions f and g in Open image in new window .
By property (d), it is easy to show that the functions of class \(\widetilde{C}^{(m+1)}(\mathbb{R}^{m})\) have absolutely convergent Fourier series. The following theorem shows that the smoothness requirement can be weakened considerably.
Theorem
If \(f\in\widetilde{C}^{r}(\mathbb{R}^{m})\) and r>m/2, then \(\sum_{n\in\mathbb{Z}^{m}}\widehat{f}(n)<+\infty\), and, therefore, \(f(x)=\sum_{n\in\mathbb{Z}^{m}}\widehat{f}(n) e^{i\langle n,x\rangle}\) for all x.
Proof
10.4.5
Now, we turn to the problem of the Fourier series representation of functions from a wider class. As already mentioned, we confine ourselves to partial sums of multiple series over rectangles. Here, we will use the notation introduced in Sect. 1.1.6. For vectors a=(a _{1},…,a _{ m }) and b=(b _{1},…,b _{ m }), the inequalities a⩽b and a<b mean that a _{1}⩽b _{1},…,a _{ m }⩽b _{ m } and a _{1}<b _{1},…,a _{ m }<b _{ m }, respectively. By a, we denote the vector (a _{1},…,a _{ m }).
The following theorem shows that the class of functions with uniformly convergent Fourier series is rather wide.
Theorem
Assume that a periodic function f satisfies the Lipschitz condition of order α, 0<α⩽1, i.e., there is an L such that f(x)−f(y)⩽L∥x−y∥^{ α } for all \(x,y\in\mathbb{R}^{m}\). Then the sums S _{ n }(f) converge uniformly to f as min{n _{1},…,n _{ m }}→+∞.
Proof
To simplify the subsequent formulas, we change the notation as follows: n=(j,k), x=(a,b) and u=(s,t). Estimating the difference Δ, we may assume, without loss of generality, that the first coordinate of the vector n does not exceed its second coordinate, i.e., j⩽k.
10.4.6
Here, we present two negative results illustrating some phenomena that may occur in the behavior of the double Fourier series and which are impossible in the onedimensional case.
The first of them is connected with the Riemann localization principle (see Theorem 10.3.3). It turns out that this principle is not true for sums over rectangles: there is a function \(f\in\widetilde{C}\) equal to zero in a neighborhood of the origin and such that the partial sums (over rectangles) of its Fourier series are unbounded at the origin. To verify this, consider a function f of the form f(s,t)=φ(s)ψ(t), where \(\varphi ,\psi\in\widetilde{C}\). It is easy to find a function φ equal to zero in the vicinity of the origin and such that S _{ j }(φ,0)≠0 for an infinite number of indices j. We take a function ψ for which the sequence \(\{S_{k}(\psi,0)\}_{k\in\mathbb{N}}\) is unbounded as the second factor (see the Schwartz example in Sect. 10.3.9). Then the product f(s,t)=φ(s)ψ(t) is equal to zero not only in the vicinity of the origin, but also in a vertical strip containing the second coordinate axis. Moreover, S _{ j,k }(f,(0,0))=S _{ j }(φ,0)S _{ k }(ψ,0). Taking an arbitrarily large j for which S _{ j }(φ,0)≠0, we can choose a k such that the sum S _{ j,k }(f,(0,0)) is larger than every preassigned number.
It can be proved that the localization principle is preserved if the usual neighborhoods of a point (a,b) are replaced by “cross neighborhoods”, i.e., by sets of the form {(s,t) min(s−a,t−b)<δ}.
The second fact that we want to mention is connected with Carleson’s theorem (see the end of Sect. 10.3.9) and is considerably harder. C.L. Fefferman^{18} observed that this theorem cannot be carried over to functions of several variables: there is a periodic function of two variables that is uniformly bounded in the square (0,2π)^{2} and whose Fourier sums over rectangles are unbounded at every point of this square. The “divergence phenomenon” manifests itself on the functions f _{ N } equal to e ^{ iNst } for 0<s,t<2π (N is a large parameter). It turns out that, despite the fact that f _{ N }=1, for every N>1 and every point (s,t), there are numbers j and k such that the quantity S _{ j,k }(f _{ N },(s,t)) is comparable with lnN. We will not discuss the Fefferman’s example in detail, but refer the reader to Exercise 10.
10.4.7
 (a)
Φ_{ n }(y)⩾0;
 (b)
∫_{ Q }Φ_{ n }(y) dy=1;
 (c)
\(\int_{Q\setminus B(\delta )}\Phi_{n}(y)\,dy\leqslant\frac{C_{\delta }}{\min\{n_{1}, \ldots,n_{m}\}}\) for every δ∈(0,π).
Thus, we can regard the functions \(\{\Phi_{n}\}_{n\in\mathbb{Z}^{m}_{+}}\) as an approximate identity with the proviso that it is now parametrized by an integer vector n and the localization property is valid as min{n _{1},…,n _{ m }}→+∞. Therefore, the following analogs of statements (b) and (c) of Theorem 10.4.1 are valid for the sums σ _{ n }(f,x).
Theorem
 (1)
If \(f\in\widetilde{C}(\mathbb{R}^{m})\), then σ _{ n }(f)⇉f on \(\mathbb{R}^{m}\) as min{n _{1},…,n _{ m }}→+∞.
 (2)
If Open image in new window for some p∈[1,+∞), then ∥σ _{ n }(f)−f∥_{ p }→0 as min{n _{1},…,n _{ m }}→+∞.
As in the onedimensional case (see the Remark in Sect. 10.4.1), the convergence of the Fejér sums in mean implies, in particular, the uniqueness theorem for multiple Fourier series:
Corollary
Functions in Open image in new window coincide almost everywhere if they have the same Fourier coefficients.
(For more general results, see Sects. 11.1.9 and 12.3.3.)
If f belongs to the class Open image in new window for some p>1, then the sums σ _{ n }(f) converge almost everywhere. Although this assumption can be weakened somewhat, it is impossible to drop it completely.
The difficulties arising in the study of the multidimensional Fejér method occur also in the “coordinatewise” generalizations of other methods, for example, of the Abel–Poisson method. We will not discuss this question in detail, instead referring the reader to the literature [Zy], vol. II, Chap. XVII.
10.4.8
In conclusion, we note that, for m>1, there are different natural ways of constructing partial sums of a multiple Fourier series and their averages. For example, instead of sums over rectangles, we could consider only sums over cubes centered at the origin and their arithmetic means. Although the corresponding kernels will not preserve sign, it can be proved that they define a generalized approximate identity satisfying the assumption (a′), less restrictive than assumption (a) (see Sect. 7.6.1).
In the twodimensional case, the arithmetic means of the “disc” Fourier sums of a periodic continuous function f converge to f uniformly (and if Open image in new window for p<+∞, then in the Open image in new window norm), but this is not the case for a larger number of variables.
EXERCISES
 1.

Let \(T(x)=\sum_{k=n}^{n} c_{k} e^{ikx}\) be a trigonometric series of order n, and let p∈[1,+∞]. Prove the Bernstein ^{19} inequality ∥T′∥_{ p }⩽2n∥T∥_{ p }. Hint. Verify that T′=−2nT∗Ψ_{ n }, where Ψ_{ n }(x)=Φ_{ n }(x) sinnx and Φ_{ n } is a Fejér kernel.
 2.

Prove that the Fejér sums cannot converge rapidly: either there is a δ>0 such that \(\f\sigma _{n}(f)\_{1}\geqslant\frac{\delta}{n}>0\) for all \(n\in\mathbb{N}\), or f≡const almost everywhere. Hint. Calculate the Fourier coefficients of the difference f−σ _{ n }(f) and apply inequality (a) of Sect. 10.3.2.
 3.

Supplement the result of Corollary 2 of Sect. 10.4.1 by proving that \(\S_{n}(f)f\_{\infty}\leqslant CL\frac{\ln n}{n}\) for α=1. Hint. To estimate the integral \(S_{n}(f,x)  f(x) = \int_{\pi}^{\pi}(f(xu)  f(x))D_{n}(u)\,du\), replace the difference f(x−u)−f(x) on each interval \([\frac{2k1}{n+1/2}\pi,\frac{2k+1}{n+1/2}\pi]\) by its value at the midpoint of this interval and then verify that the integral of the Dirichlet kernel over the interval admits an estimate O(1/k ^{2}).
 4.

Prove that the Fourier cosine coefficients can tend to zero arbitrarily slowly, i.e., for every sequence {c _{ n }}_{ n⩾1} decreasing to zero, there is a function Open image in new window such that c _{ n }⩽a _{ n }(f) for all \(n\in\mathbb{N}\). Hint. Dominate {c _{ n }}_{ n⩾1} by a convex sequence and apply Theorem 10.4.2.
 5.

Let the sequence of coefficients of a series \(\sum_{n=1}^{\infty} c_{n} \cos nx\) be convex. Prove that the Open image in new window norms of the partial sums are bounded if and only if c _{ n }=O(1/lnn); prove that the given series converges in Open image in new window if and only if c _{ n }=o(1/lnn) as n→∞.
 6.

Let \(S(x)=\sum_{n=1}^{\infty} c_{n}\sin nx\) where c _{ n }↓0. Prove that the boundedness and the continuity of the function S is equivalent to the relation \(c_{n}=O (\frac{1}{n} )\) and \(c_{n}=o (\frac{1}{n} )\), respectively.
 7.

Prove that the following version of Parseval’s identity is valid for functions Open image in new window and Open image in new window : the series \(\sum_{n\in\mathbb{Z}}\widehat{f}(n)\overline{\widehat {g}(n)}\) Cesaro converges to \(\frac{1}{2\pi}\int_{\pi}^{\pi} f(x)\overline{g(x)}\,dx\).
 8.
 Prove that the Fourier series of every function f in Open image in new window can be integrated termwise over every rectangular parallelepiped P,(the sum of the series on the righthand side of the equation is regarded as the limit of the partial sums over rectangles).$$\int_Pf(x)\,dx=\sum_{n\in\mathbb{Z}^m} \widehat{f}(n) \int_Pe^{i\langle n,x\rangle}\,dx $$
 9.

Let a function Open image in new window be such that \(\widehat{f}(n)\geqslant0\) for all \(n\in\mathbb{Z}^{m}\). Prove that if f is bounded and continuous at the origin, then its Fourier series converges absolutely (therefore, f coincides with a function of class \(\widetilde{C}\) almost everywhere). Hint. Use the fact that the sums σ _{ n }(f,0) are bounded.
 10.
 To construct a continuous function of two variables for which the Fourier series diverges everywhere in the square (0,2π)^{2} (see Sect. 10.4.6), prove that the Fourier sums S _{ j,k }(f _{ N }) of the function f _{ N }(s,t)=e ^{ iNst } (N⩾1, 0<s,t<2π) over the rectangles [−j,j]×[−k,k] satisfy the following inequalities at each point of the given square (the constants at the Oterms depend only on s and t):
 (a)
S _{ j,k }(f _{ N };s,t)=O(lnj);
 (b)
if k>2πN, then \(S_{j,k}(f_{N};s,t)= O (1+\frac{\ln j}{k2\pi N} )\);
 (c)
\(S_{j,k}(f_{N};s,t)\geqslant\frac{1}{2\pi}\ln N+O(1)\) for j=[Ns] and k=[Nt].
Conclude from this that, for a sufficiently small r>0 and N _{ n }=e ^{ n!}, the Fourier series of the function \(F(s,t)=\sum_{n=1}^{\infty} r^{n} e^{iN_{n}st}\) diverges unboundedly (the sums S _{ j,k }(F) are unbounded) at each point of the square (0,2π)^{2}.
 (a)
10.5 The Fourier Transform
10.5.1
We introduce one of the main concepts of this chapter.
Definition
An important property of the Fourier transform relates the operations of convolution and multiplication.
Theorem
If Open image in new window , then \(\widehat{f*g}(y)=\widehat{f}(y)\,\widehat{g}(y)\) \((y\in\mathbb{R}^{m})\). Moreover, \(\int_{\mathbb{R}^{m}}\widehat{f}(y)\,g(y)\,dy= \int_{\mathbb{R}^{m}}f(y)\,\widehat{g}(y)\,dy\).
Proof
The second relation is proved similarly. □
We consider some examples.
Example 1
Example 2
Example 3
Example 4
Example 5
Let a,u>0, and let f(x)=x ^{ a−1} e ^{−ux } for x>0 and f(x)=0 for x<0. Then \(\widehat{f}(y)=\frac{\Gamma(a)}{(u+2\pi iy)^{a}}\) (we use the branch of the power function z ^{ a } equal to 1 at z=1). This was established in Example 1 of Sect. 7.1.7.
Before passing to a more detailed study of the properties of the Fourier transform, we show the usefulness of this notion by one more example.
Example 6
10.5.2
We establish elementary relations between differentiation and the Fourier transform.
Theorem
 (1)if the partial derivative \(g=\frac{\partial f}{\partial x_{k}}\) is summable and continuous for some k=1,…,m, then$$\widehat{g}(y)=2\pi iy_k \widehat{f}(y)\quad\bigl(y\in \mathbb{R}^m\bigr); $$
 (2)if the product ∥x∥f(x) is summable, then \(\widehat{f}\in C^{1}(\mathbb{R}^{m})\) and the equation$$\frac{\partial\widehat{f}(y)}{\partial y_k}=2\pi i\widehat {f}_k(y),\quad {\textit{where}}\ f_k(x)=x_kf(x)\ \bigl(x\in\mathbb{R}^m \bigr), $$
Proof
To obtain the equation of (2), we must apply the Leibnitz rule (see Sect. 7.1.5). The functions f _{1},…,f _{ m } are summable by assumption. Therefore, their Fourier transforms and the first order partial derivatives of \(\widehat{f}\) are continuous everywhere. Consequently, \(\widehat{f}\in C^{1}(\mathbb{R}^{m})\). □
Corollary
If Open image in new window is a compactly supported function, then \(\widehat{f}\in C^{\infty}(\mathbb{R}^{m})\); if \(f\in C_{0}^{\infty}(\mathbb {R}^{m})\), then the product \(\y\^{p}\widehat{f}(y)\) is summable in \(\mathbb{R}^{m}\) for every p>0.
Proof
The fact that \(\widehat{f}\) is infinitely differentiable follows directly from the second assertion of the theorem because the product ∥x∥^{ n } f(x) is summable for every \(n\in\mathbb{N}\).
In many problems, it is important to know the rate of decrease of the Fourier transform at infinity. The theorem proved above shows that a fast decrease can be provided by the smoothness of the function in question. How accurate are these conditions? What can be expected if smoothness fails on a “small” set? The following examples are devoted to such results.
Example 1
It can be proved that \(\widehat{f}(y)>0\) for 0<p⩽2 (for 0<p⩽1, this follows from the result of Example 2 of Sect. 4.6.6 and the fact that the function \(e^{x^{p}}\) is convex).
Example 2
Example 3
It follows from the theorem that the condition f(x)=O(∥x∥^{−p }) as ∥x∥→+∞ implies the smoothness of the Fourier transform for p>m+1. It turns out that this restriction cannot be weakened essentially. To verify this, we show that if f(x)∼∥x∥^{−p } as ∥x∥→+∞, then the differentiability of \(\widehat{f}\) at zero implies the inequality p>m+1.
Without loss of generality, we may assume that f⩾0. Indeed, we know that f(x)⩾0 for large ∥x∥, but changing the function on an arbitrary ball (for example, putting f(x)=0 inside the ball), we change the Fourier transform of f by an infinity differentiable function.
10.5.3
There is an obvious analogy between the expansion of a periodic function in a Fourier series and the Fourier integral representation of a nonperiodic function. The following theorem shows that these two problems share not only some superficial analogies but are connected in essence. To show this, we need the following easy lemma.
Lemma
Proof
Theorem
From the theorem, it obviously follows that the convergence tests for Fourier series, obtained in Sect. 10.3.4, can be carried over to the Fourier integrals. In particular, the inversion formula is valid at a point x if Dini’s condition is fulfilled at x with C=f(x). We leave it to the reader to state an analog of the Dirichlet–Jordan test.
Proof
Now, we once again turn to Examples 2 and 3 considered in Sect. 10.5.1.
Example 1
Example 2
10.5.4
Generalizing the inversion formula to functions of several variables, we confine ourselves to the most important case where the Fourier transform is summable. In this connection, we note that Dini’s condition providing the validity of the inversion formula in the onedimensional case is a local property of a summable function, whereas the summability of the Fourier transform is a global property.
In contrast to the onedimensional setting, now, when deriving the inversion formula, we cannot use the equiconvergence of the expansions in the Fourier series or Fourier integral since Theorem 10.5.3 cannot be carried over to the multidimensional case (see Exercise 6).
Theorem
Let Open image in new window . If Open image in new window , then inversion formula (5) is valid for almost all x in \(\mathbb{R}^{m}\).
We remark that the righthand side of Eq. (5) continuously depends on x since \(\widehat{f}\) is summable. Therefore, the condition of the theorem (the summability of \(\widehat{f}\)) can be fulfilled only if the function f is equivalent to a continuous function. Moreover, Eq. (5) is valid at all points where f is continuous because it is valid on a set of full measure. In particular, if f is continuous and its Fourier transform is summable, then Open image in new window for all \(x\in\mathbb{R}^{m}\).
Proof
Now, we can finish the proof, referring to Theorem 10.3.4, from which it follows that the limit on the lefthand side of Eq. (7) coincides with f(x) almost everywhere. However, it is possible to dispense with the use of the theorem based on the notion of a Lebesgue point and on Theorem 4.9.2 on differentiation of an integral with respect to a set. We show that the lefthand side of Eq. (7) tends to f(x) almost everywhere as t tends to zero along a sequence.
Indeed, let {t _{ n }} be a sequence such that \(t_{n}\underset{n\to\infty}{\longrightarrow}0\). Theorem 9.3.3 implies that \(f*W_{t_{n}}\underset{n\to\infty}{\longrightarrow} f\) in mean, and, consequently, in measure (see Theorem 9.1.2). By Riesz’s theorem (see Sect. 3.3.4), there is a subsequence \(\{t_{n_{k}}\}\) of {t _{ n }} such that, almost everywhere \(f*W_{t_{n_{k}}}\to f\) as k→∞. Replacing t by \(t_{n_{k}}\) in Eq. (7) and passing to the limit, we obtain the required result. □
Example
The summability of the Fourier transform is important in many problems (see, for example, Sect. 10.6.4). The result of Exercise 7 shows that this condition is necessarily fulfilled if \(\widehat{f}\geqslant0\) and the function f is continuous (or at least bounded in a neighborhood of zero). In this connection, we recall (see Example 2 of Sect. 4.6.6) that \(\widehat{f}\geqslant0\) if f is an even function summable on \(\mathbb{R}\) and convex on (0,+∞). Together with the inversion formula, this proves the following statement.
Corollary
If an even continuous function f is summable on the real line and is convex on the positive semiaxis, then f is the Fourier transform of a nonnegative summable function.
The fact just proved remains valid even if, instead of the summability of f, we assume only that \(f(x)\underset{x\to+\infty}{\longrightarrow}0\), but, in this case, the proof invokes a subtler reasoning (see [Luk], Pólya’s theorem).
10.5.5
Here, we discuss one more important property of the Fourier transform, its injectivity on the entire set of summable functions. Of course, there is no injectivity in the literal sense because distinct equivalent (i.e., coinciding almost everywhere) functions have the same Fourier transform. However, Theorem 10.5.4 shows that the injectivity holds up to equivalence on the set of functions with summable Fourier transform. To strengthen this result, we generalize Definition 10.5.1 somewhat.
Definition
Let μ be a finite Borel measure on \(\mathbb {R}^{m}\). The function \(y\mapsto\widehat{\mu}(y)\equiv\int_{\mathbb{R}^{m}}e^{2\pi i\langle y,x\rangle}\,d\mu (x)\) is called the Fourier transform of μ.
If a measure μ has a density f with respect to Lebesgue measure, then \(\widehat{\mu}=\widehat{f}\).
Now, we establish an important result connected with the injectivity of the Fourier transform of a measure.
Theorem
If two finite Borel measures μ and ν have the same Fourier transform, then the measures coincide.
Proof
It follows from the above theorem that the Fourier transform is injective up to equivalence on the set of summable functions.
Corollary 1
If two summable functions f and g have the same Fourier transform, they coincide almost everywhere.
Proof
It is clear that the Fourier transform of the functions \(\overline{f} \) and \(\overline{g}\) also coincide. Consequently, the functions \(\mathcal{R}e\,f=(f+\overline{f})/2\) and \(\mathcal{R}e\,g=(g+\overline{g})/2\), as well as the imaginary parts of the functions f and g, have the same Fourier transform. Therefore, we may assume that the functions f and g are real.
If they are nonnegative, the theorem just proved implies that the measures with the densities f and g coincide. It was proved in Sect. 4.5.4 that, in this case, the densities coincide almost everywhere.
Corollary 2
If finite Borel measures μ and ν have the same values on all halfspaces (in \(\mathbb{R}^{m}\)), then they coincide.
Proof
10.5.6
Using the results of the previous section, we will prove here that the system of Hermite polynomials is complete. The method we use enables us to consider a more general situation and prove that the family of monomials in m variables, i.e., the products \(x^{n}=x_{1}^{n_{1}}\cdots x_{m}^{n_{m}}\), where \(x=(x_{1},\ldots,x_{m})\in\mathbb{R}^{m}\) and \(n=(n_{1},\ldots, n_{m})\in\mathbb{Z}^{m}_{+} \), is complete in Open image in new window for a wider class of measures.
Theorem
If a Borel measure μ on \(\mathbb{R}^{m}\) satisfies the condition \(\int_{\mathbb{R}^{m}}e^{a\x\}\,d\mu(x)<+\infty\) for some a>0, then the family of all monomials is complete in the space Open image in new window .
Proof
Corollary
The Hermite polynomials are complete in Open image in new window with \(d\mu(x)=e^{x^{2}}\,dx\).
This is a special case of the theorem for m=1. We also remark that the theorem implies that the Laguerre functions are complete (for the definition, see Exercise 3 of Sect. 10.2).
The following example shows that the result obtained in the theorem is quite sharp.
Example
We verify that the polynomials are not complete in the space Open image in new window with measure μ having density \(e^{x^{p}}\) (0<p<1) with respect to the onedimensional Lebesgue measure (for p⩾1 this effect is ruled out by the theorem just proved).
10.5.7
The present and two following sections are devoted to an important theorem, due to Plancherel,^{20} and its corollaries. The traditional formulation of the theorem would require us to invoke some concepts from functional analysis and operator theory. To avoid this, we first establish an analytic fact constituting the core of the theorem.
Theorem
(Plancherel)
If Open image in new window , then Open image in new window and \(\\widehat{f}\_{2} = \f\_{2}\).
Proof
Let {ω _{ t }}_{ t>0} be a Sobolev approximate identity in \(\mathbb{R}^{m}\) (see Sect. 7.6.2) and f _{ t }=f∗ω _{ t }.
The concluding part of the proof can be somewhat shortened. Indeed, since \(\widehat{\omega}_{t}\leqslant\int_{\mathbb{R}^{m}} \omega(x)\,dx=1\), we have \(\widehat{f}_{t}\leqslant\widehat{f}\). Since \(\widehat{f}_{t}\underset{t\to0}{\longrightarrow}\widehat{f}\), we can pass to the limit on the righthand side of Eq. (8) by the generalization of B. Levi’s theorem given in Exercise 4 of Sect. 4.8.
10.5.8
We show how Plancherel’s theorem can be used to generalize the concept of the Fourier transform to functions in Open image in new window .
Lemma
Let Open image in new window . If {f _{ n }}_{ n⩾1} is a sequence of functions in Open image in new window convergent to f in the Open image in new window norm, then the sequence \(\{\widehat{f}_{n}\}_{n\geqslant1}\) also converges in the Open image in new window norm. Its limit does not depend (up to equivalence) on the choice of the sequence {f _{ n }}_{ n⩾1}.
Proof
The lemma just proved allows us to extend the definition of the Fourier transform to the functions in Open image in new window .
Definition
By the Fourier transform of a function Open image in new window , we mean the limit in the Open image in new window norm of the functions \(\widehat{f}_{n}\), where {f _{ n }}_{ n⩾1} is an arbitrary sequence of functions in Open image in new window such that \(\f_{n}f\_{2}\underset{n\to\infty}{\longrightarrow}0\).
Thus, the Fourier transform of a function in Open image in new window is also squaresummable. As before, we will denote the Fourier transform of f by \(\widehat{f}\). However, one must keep in mind that now the Fourier transform is defined up to equivalence and the symbol \(\widehat{f}\) refers to many functions. If f is summable, then, among these functions, is the Fourier transform defined in Sect. 10.5.1. For definiteness, the latter is sometimes called the classical Fourier transform. What has just been said also applies to the inverse transform, which, as before, is denoted by Open image in new window .
Elementary properties of the Fourier transform of squaresummable functions can be obtained from the properties of the classical Fourier transform by a passage to the limit.
Theorem
 (1)
\(\\widehat{f}\_{2}=\f\_{2}\);
 (2)
if Open image in new window and \(\f_{n}f\_{2} \underset{n\to\infty}{\longrightarrow}0\), then \(\\widehat{f}_{n}\widehat{f}\_{2} \underset{n\to\infty}{\longrightarrow}0\), and a similar statement holds for the inverse transform;
 (3)
we have Open image in new window ;
 (4)
\(\langle\widehat{f},\widehat{g}\rangle=\langle f,g\rangle\) for every function Open image in new window . In particular, the Fourier transform preserves orthogonality: if f⊥g, then \(\widehat{f}\perp\widehat{g}\).
Proof
Let \(\{\varphi _{n}\}_{n\geqslant1}\in C_{0}^{\infty}(\mathbb{R}^{m})\) be a sequence of functions converging to f in the Open image in new window norm. It is obvious that these functions and their Fourier transforms belong to Open image in new window .
(1) It is clear that \(\\widehat{\varphi }_{n}\widehat{f}\_{2}\underset{n\to\infty}{\longrightarrow}0\) by the definition of \(\widehat{f}\). By Plancherel’s theorem, we have \(\\widehat{\varphi }_{n}\_{2}=\\varphi _{n}\_{2}\). Therefore, it is sufficient for us to use the continuity of the norm and pass to the limit in this equation.
(2) Obviously, \(\\widehat{f}_{n}\widehat{f}\_{2}=\\widehat{f_{n}f}\_{2}=\ f_{n}f\_{2} \underset{n\to\infty}{\longrightarrow}0\).
(3) We will prove only the equality Open image in new window (the other one is proved similarly). Since \(\varphi _{n}\underset{n\to\infty}{\longrightarrow} f\), we obtain by definition that \(\widehat{\varphi }_{n}\underset{n\to\infty}{\longrightarrow}\widehat{f}\), and, by property 2) applied to the inverse transform, we have Open image in new window Open image in new window . At the same time, Open image in new window by Theorem 10.5.4. Thus, it only remains to pass to the limit (in the Open image in new window norm) in the last equality.
(4) For the proof, we must use the identity \(4f\overline{g}=f+g^{2}+f+ig^{2} fg^{2}fig^{2}\) and apply relation (1) to the functions f±g and f±ig. □
10.5.9
Plancherel’s theorem implies an inequality known as the uncertainty principle. Without touching on its physical meaning (the impossibility of simultaneously determining the exact values of the coordinates and impulse of a quantum object), we mention only its consequence: if f≠0 only in the vicinity of the origin, then the quantity \(\widehat{f}\) is not small at some remote points (the Fourier transform “blurs”). In the onedimensional case, the reader can see this effect in the example of functions \(\frac{1}{2t}\chi_{(t,t)}\) forming an approximate identity.
In the precise formulation of the uncertainty principle, we confine ourselves to infinitely differentiable compactly supported functions of one variable (more general statements are given in Exercises 10 and 11).
Theorem
Proof
10.5.10
EXERCISES
 1.

Find the Fourier transform of the product e ^{2πi〈h,x〉} f(x), where Open image in new window and \(h\in\mathbb{R}^{m}\).
 2.

Let Open image in new window and \(f_{E}(x)=\frac{1}{\lambda _{m}(E)}\int_{E}f(x+t)\,dt\), where \(E\subset\mathbb{R}^{m}\) is a set of finite positive measure. Prove that \(\widehat{f}_{E}\leqslant\widehat{f}\).
 3.
 Let a function f in Open image in new window vanish outside the cube (−π,π)^{ m }, and let \(F(t)=\widehat{f}(t/2\pi)\). Prove that(the sum on the righthand side of the equation is understood as the limit of the partial sums over rectangles).$$F(t)=\sum_{n\in\mathbb{Z}^m}F(n)\prod _{j=1}^m\frac{\sin\pi(t_jn_j)}{\pi (t_jn_j)} $$
 4.
 A function f defined on \(\mathbb{R}^{m}\) is called positive definite iffor all \(n\in\mathbb{N}\), \(x_{j}\in\mathbb{R}^{m}\) and \(z_{j}\in\mathbb{C}\). Prove that the Fourier transform of a finite Borel measure μ is a positive definite function.$$\sum_{1\leqslant j,k\leqslant n}f(x_jx_k)z_j \overline{z}_k\geqslant0 $$
 5.

Prove that if g is the generalized derivative of f with respect to the kth coordinate (see Exercise 5 of Sect. 9.3), then the relation \(\widehat {g}(y)=2\pi iy_{k}\widehat{f}(y) \) of statement (1) of Theorem 10.5.2 remains valid.
 6.

Verify that, in general, the equiconvergence of the expansions in Fourier series and Fourier integrals does not take place in the multidimensional case. Hint. Use the same idea as in the first part of Sect. 10.4.6.
 7.

Let a function Open image in new window be bounded in a neighborhood of zero. Prove that if \(\widehat{f} \geqslant0\), then Open image in new window . Hint. By Eq. (7), prove that the integrals \(\int_{\mathbb{R}^{m}}e^{\pi t^{2}\u\^{2}}\widehat{f}(u)\,du\) are bounded for t>0 and apply Fatou’s theorem.
 8.
 Let a measure μ on \(\mathbb{R}^{m}\) be such that \(\int_{\mathbb{R}^{m}}e^{a\x\}\,d\mu(x)<+\infty\) for some a>0. Generalizing Theorem 10.5.6, prove that, for p>1, every function Open image in new window satisfying the conditionis equal to zero μalmost everywhere.$$\int_{\mathbb{R}^m}f(x)x^n\,d\mu(x)=0\quad\text{for all}\ n\in \mathbb{Z}^m_+ $$
 9.

Let φ _{1},φ _{2},… be functions in Open image in new window . Prove that the systems \(\{\varphi _{n}\}_{n\in\mathbb{N}}\) and \(\{\widehat{\varphi }_{n}\} _{n\in\mathbb{N}}\) are complete or not simultaneously.
 10.
 Let Open image in new window and ∥f∥_{2}=1. Prove that the inequalityis valid for all \(a,b\in\mathbb{R}^{m}\).$$\int_{\mathbb{R}^m}\xa\^2 \bigf(x)\big^2\,dx \cdot \int_{\mathbb{R}^m}\yb\^2\big\widehat{f}(y)\big^2 \,dy\geqslant\frac{m^2}{16\pi^2} $$
 11.
 Assume that the values of a function Open image in new window are small outside a ball B(a,r) in the sense thatand the values of \(\widehat{f}\) are small (in the same sense) outside a ball B(b,R). Prove that \(rR>\frac{m}{8\pi}\).$$\int_{\mathbb{R}^m\setminus B(a,r)}\xa\^2\bigf(x)\big^2\,dx< \frac{1}{2} \int_{\mathbb{R}^m}\xa\^2\bigf(x)\big^2\,dx, $$
 12.

Supplement the result of Example 3 of Sect. 10.5.2 by proving that if \(f(x)\underset{\x\\to+\infty}{\sim}\x\^{pm}\), as ∥x∥→+∞ for some p∈(0,2) and the function f is even, then \(\widehat{f}(0)\widehat{f}(y)\underset{y\to0}{\sim} C_{p}\y\^{p}\); for p=2, this relation must be replaced by \(\widehat{f}(0)\widehat{f}(y)\underset{y \to0}{\sim} C\y\^{2}\ln\frac{1}{\y\}\). The coefficients C _{ p } and C depending on the dimension can be expressed in terms of the gamma function.
 13.

Let P, Q be algebraic polynomials with deg Q>deg P. Show that the Fourier transform of the fraction \(f=\frac{P}{Q}\) vanishes on the negative halfaxis (\(\widehat{f}(y)=0\) for y⩽0) if all roots of the denominator Q lie in the lower halfplane (i.e., their imaginary parts are negative).
10.6 ^{⋆}The Poisson Summation Formula
In this section, by the periodicity of a function of several variables, we mean 1periodicity with respect to each variable. Speaking of the Fourier series of a periodic function summable on the cube \((\frac{1}{2},\frac{1}{2} )^{m}\), we mean a series in the system of exponential functions \(\{e^{2\pi i\langle n,x\rangle} \}_{n\in\mathbb{Z}^{m}}\).
10.6.1
We need the following statement.
Lemma
 (a)
series (2) converges absolutely for almost all x;
 (b)
its sum S is a 1periodic function; this function is summable on \((\frac{1}{2},\frac{1}{2})\), and Eq. (2) can be integrated termwise;
 (c)the Fourier coefficients of the function S are equal to the values of \(\widehat{f}\) at the integer points,$$\int_{\frac{1}{2}}^{\frac{1}{2}} S(x) e^{2\pi inx}\,dx=\widehat {f}(n)\quad (n\in\mathbb{Z}). $$
Proof
Since the 1periodicity of the function S is obvious, statement (b) follows from the inequality S⩽F. Since F also dominates all partial sums of series (2), the series can be integrated termwise.
Example 1
Example 2
Example 3
Without any assumptions on the function f (except the summability), Eq. (3) is valid in the following “weak” sense: after integrating both sides of the equation over an arbitrary interval, we obtain convergent series with equal sums. This follows immediately from our ability to integrate Fourier series termwise (Theorem 1 of Sect. 10.3.6).
We also remark that not only does (1) follow from (3), but also (3) can be regarded as Eq. (1) for the shift f _{−x } of the function f since \(\widehat{f}_{x}(n)=\widehat{f}(n) e^{2\pi inx}\) (see the beginning of Sect. 10.5.1).
To derive Eq. (1) from (3), we must be sure that the latter is valid for x=0. For this, it is not sufficient, for example, that series (3) converges almost everywhere. Therefore, of particular interest to us is to find conditions under which the function S has a Fourier series expansion everywhere. One such conditions is given in Exercise 2. Other versions of sufficient conditions for functions of several variables will be established in the next section.
10.6.2
We give two types of conditions under which Eq. (5) is valid.
Theorem 1
Let f be a continuous function on \(\mathbb {R}^{m}\) such that f(x)=O(∥x∥^{−p }) and \(\widehat{f}(x)=O(\x\^{p})\) as ∥x∥→+∞ for some p>m. Then Eq. (5) is valid for all \(x\in \mathbb{R}^{m}\).
Proof
First, we observe that Open image in new window since the summability on an arbitrary ball follows from the continuity of f and the summability outside the ball follows from the estimate f(x)=O(∥x∥^{−p }).
Under our assumptions, the series on both sides of Eq. (5) converge absolutely and uniformly on every ball, and, since their terms are continuous, the sums of the series are also continuous. Thus, the righthand side of (5) is a uniformly convergent Fourier series of the sum on the lefthand side of (5). □
It is not difficult to give an interpretation of the sum of a multiple series in the case of absolute convergence. Otherwise, it is necessary to clarify the definition of this sum. First of all, this concerns the series on the righthand side of Eq. (5) (the series on the lefthand side of (5) converges absolutely for almost all x). The following theorem enables us to consider the situation in which the series on the righthand side of (5) does not converge absolutely (see also Exercises 3 and 4).
Theorem 2
Let a function f satisfy the Lipschitz condition with exponent α in \(\mathbb{R}^{m}\), i.e., there is a positive L such that f(x)−f(x′)⩽L∥x−x′∥^{ α } for all \(x,x'\in\mathbb{R}^{m}\). We assume, in addition, that the function decreases rapidly at infinity, i.e., f(x)=O(∥x∥^{−p }) as ∥x∥→+∞ for some p>m. Then Eq. (5) is valid for all \(x\in \mathbb{R}^{m}\) (the sum on the righthand side of (5) is understood as the limit of rectangular partial sums).
Proof
Corollary
If a function f having bounded first order derivatives satisfies the condition f(x)=O(∥x∥^{−p }) as ∥x∥→+∞ for some p>m, then Eq. (5) holds for every point x (the sum on the righthand side of the equation is understood as the limit over rectangular partial sums).
10.6.3
The Poisson summation formula has proved to be an effective tool for solving various problems (see Sects. 10.6.4 and 10.6.5). However, before passing to these technically more involved applications, we use the summation formula to supplement the uncertainty principle established in Sect. 10.5.9, according to which the functions f and \(\widehat{f}\) cannot be concentrated on “small sets” simultaneously (see also Exercises 10 and 11 of Sect. 10.5). The following statement is valid (see [B]).
Theorem
Let a summable function f on \(\mathbb{R}^{m}\) be such that the sets \(A = \{x \in\mathbb{R}^{m}f(x)\ne0\}\) and \(B=\{y\in\mathbb {R}^{m}\,\widehat{f}(y)\ne0\}\) have finite measures, then f(x)=0 almost everywhere.
Proof
Since the kth Fourier coefficient of the function S _{ y } is equal to \(\widehat{f_{y}} (k)\) (see statement (c)) of Lemma 10.6.1), we obtain that S _{ y } coincides with a trigonometric polynomial almost everywhere. The set E _{ y }={x∈Q  S _{ y }(x)≠0} is contained in the union ⋃_{ n }(−n+A)∩Q, and, therefore, λ _{ m }(E _{ y })⩽λ _{ m }(A)<1=λ _{ m }(Q). Since a nonzero trigonometric polynomial does not vanish almost everywhere, we obtain that S _{ y }(x)=0 almost everywhere. Consequently, \(0=\widehat{S_{y}}(0)=\widehat{f_{y}}(0)=\int_{\mathbb{R}^{m}}f_{y}(x)\, dx=\widehat{f}(y)\) for almost all y. By the uniqueness theorem, f=0 almost everywhere. □
We remark that the proof of the theorem does not use Eq. (5). It is based only on statement (c) of Lemma 10.6.1 (more precisely, on its mdimensional modification).
10.6.4
When is the family {ω _{ ε }}_{ ε>0} an approximate identity as ε→0? In Sect. 10.4.3, we obtained a sufficient condition. Now, using the Poisson summation formula, we can a give much more complete answer to this question. It turns out that it is sufficient that the Fourier transform \(\widehat{M}\) be summable (as follows from the result of Exercise 6, this condition is also necessary).
Example
10.6.5
An interesting application of the Poisson identity is in one of the solutions to the Gauss problem on determining the number N _{ m }(R) of points of the integer lattice \(\mathbb{Z}^{m}\) that lie in the closed ball of a large radius R. The number N _{ m }(R) is close to the volume of the ball, \(N_{m}(R)\lambda _{m}(\overline{B}(R))=O(R^{m1})\) as R→+∞. This can easily be verified by considering the unit cubes centered at the lattice points, i.e., the cubes \(n+[\frac{1}{2},\frac{1}{2}]^{m}\), \(n \in\mathbb{Z}^{m}\). Since, for \(n \in\overline{B}(R)\), their union contains the ball \(B(R\sqrt{m})\) and is contained in the ball \(\overline{B}(R+\sqrt{m})\), we see that the number N _{ m }(R), being equal to the volume of the union of these cubes, lies between \(\lambda _{m}(B(R\sqrt{m}))\) and \(\lambda _{m}(B(R+\sqrt{m}))\). In other words, \(\alpha _{m}(R  \sqrt{m})^{m} \leqslant N_{m}(R) \leqslant \alpha _{m}(R + \sqrt{m})^{m}\), where α _{ m } is the volume of the unit cube in \(\mathbb{R}^{m} \).
Much greater efforts are needed to sharpen this elementary estimate. However, we first observe that the exponent θ in the relation N _{ m }(R)=α _{ m } R ^{ m }+O(R ^{ θ }) cannot be less than m−2. Indeed, the function R↦N _{ m }(R) makes a jump at \(R^{2}\in\mathbb{N}\), the value of which is equal to the number of lattice points on the sphere of radius R. Since the principal term α _{ m } R ^{ m } of the asymptotic depends continuously on R, the exponent θ must be so large that the number of points on the sphere of radius R be dominated by a summand proportional to R ^{ θ }. The number of lattice points in the spherical layer R<∥x∥<2R is of order R ^{ m }. The number of spheres containing these points is at most 3R ^{2} (every such sphere is defined by the equation ∥x∥^{2}=t with an integer parameter t lying between R ^{2} and (2R)^{2}). At least one of the spheres contains at least const R ^{ m−2} points. Therefore, necessarily θ⩾m−2.
10.6.6
What can be said about the exactness of formula (10)? Since \(\theta= m  2+\frac{2}{m+1}\), we see that, for large m, its error is close to the minimum possible value O(R ^{ m−2}). As we know, for m>4, the best estimate is achieved, namely, relation (10) is valid with θ=m−2 (for m=4, this relation is valid for every θ>2). For m=3 the minimum value of the exponent θ is still unknown. As we have verified in the previous section, it does not exceed \(\frac{3}{2}\) and is not less than 1. There is a conjecture stating that the exponent θ can be taken arbitrarily close to 1, but it has only been proved that θ⩽29/22. For the history of the problem, see the paper [CI] or the book [LK].
We consider the twodimensional case in more detail. We have proved that, for m=2, formula (10) is valid with θ=2/3. In the study of the Gauss problem, this is the first nontrivial result, which was obtained by Sierpiński in 1906 and has been sharpened several times since then. More sophisticated methods made it possible to decrease the exponent θ to \(\frac{131}{208}\), but it is still unknown whether the value of θ can be taken arbitrarily close to \(\frac{1}{2}\). This bound cannot be lowered. As Hardy and Landau^{21} proved independently in 1915, the relation N _{2}(R)=πR ^{2}+O(R ^{ θ }) holds only for \(\theta\geqslant\frac{1}{2}\). We give the proof of this result, based on the paper [EF] (see also Exercise 9).
10.6.7
However, if m=4l+1, then the superior limit of the quotient \(\frac{1}{R^{m1}}\int_{[0,1]^{m}} \Delta_{R}(t) ^{2}\,dt\) as R→+∞ is positive and the inferior limit is zero.
EXERCISES
 1.

Supplement the statement of Lemma 10.6.1 by proving that the series \(\sum_{n\in\mathbb{Z}}f (x+n)\) converges to S(x) not only almost everywhere, but also in mean.
 2.

Let an absolutely continuous function f and its derivative be summable on \(\mathbb{R}\). Prove that Eq. (3) is valid for all \(x\in\mathbb{R}\).
 3.

Verify that, in Theorem 2 of Sect. 10.6.2, the Lipschitz condition can be weakened to the assumption that the inequality f(x+h)−f(x)⩽R ^{ q }∥h∥^{ α }, where ∥h∥⩽1 and ∥x∥⩽R (here, q is a fixed nonnegative number), holds for every R>1.
 4.

Using the result of the previous exercise, show that, in Corollary 10.6.2, the assumption that the derivatives are bounded can be replaced by the requirement that they are dominated by a polynomial.
 5.

Let Open image in new window and \(\widehat {f}\geqslant0 \) everywhere. Prove that Eq. (5) is valid at every point \(x\in\mathbb{R}^{m}\) (the series on the righthand side of (5) converges absolutely).
 6.

Prove that the condition Open image in new window is not only sufficient but also necessary for the boundedness of the Open image in new window norms of the sums ω _{ ε } (see formula (7)).
 7.

Prove that the function ω _{ ε } in the example of Sect. 10.6.4 admits the estimate \(\omega_{\varepsilon }(x)=O(\varepsilon ) (1 + (\varepsilon ^{2} +4\pi^{2}\x\^{2})^{\frac{m+1}{2}} )\) (the constant in the Oterm depends only on the dimension) and, therefore, ω _{ ε } is dominated by a summable “humpshaped” majorant.
 8.

Prove that, as ε→0, the functions \(\omega_{\varepsilon }(x) = \sum_{n\in\mathbb{Z}^{m}} e^{\varepsilon \n\^{2}}e^{2\pi i\langle n,x \rangle}\) form an approximate identity having the strong localization property and a “humpshaped” majorant.
 9.

Verify that the reasoning of Sect. 10.6.6 can be used to obtain the following stronger result (see [EF]): the fraction \(\Delta(R) /\sqrt{\frac{R}{\ln R}}\) does not tend to zero as R→+∞.
Footnotes
 1.
Pythagoras (Πυϑαγóρας) (circa 570–500 BC)—Greek philosopher and mathematician.
 2.
Friedrich Wilhelm Bessel (1784–1846)—German mathematician.
 3.
Jean Baptiste Joseph Fourier (1768–1830)—French mathematician.
 4.
Ernest Sigismund Fisher (1875–1954)—German mathematician.
 5.
MarcAntoine Parseval (1755–1836)—French mathematician.
 6.
Fedor L’vovich Nazarov (born 1967)—Russian mathematician.
 7.
Adolf Hurwitz (1859–1919)—German mathematician.
 8.
Charles Hermite (1822–1901)—French mathematician.
 9.
Joseph Leonard Walsh (1895–1973)—American mathematician.
 10.
Andrei Nikolaevich Kolmogorov (1903–1987)—Russian mathematician.
 11.
Edmond Nicolas Laguerre (1834–1886)—French mathematician.
 12.
Ulisse Dini (1845–1918)—Italian mathematician.
 13.
Marie Ennemond Camille Jordan (1838–1922)—French mathematician.
 14.
Lennart Axel Edvard Carleson (born 1928)—Swedish mathematician.
 15.
Arnaud Denjoy (1884–1974)—French mathematician.
 16.
Ernesto Cesaro (1859–1906)—Italian mathematician.
 17.
Lipót Fejér (1880–1959)—Hungarian mathematician.
 18.
Charles Louis Fefferman (born 1949)—American mathematician.
 19.
Sergei Natanovich Bernstein (1880–1968)—Russian mathematician.
 20.
Michel Plancherel (1885–1967)—Swiss mathematician.
 21.
Edmund Georg Hermann Landau (1877–1938)—German mathematician.
References
 [AIN1]Alimov, Sh.A., Il’in, V.A., Nikishin, E.M.: Questions on the convergence of multiple trigonometric series and spectral expansions. I. Usp. Mat. Nauk 31(6), 28–83 (1976). 10.4.8 zbMATHGoogle Scholar
 [AIN2]Alimov, A.Sh.A., Il’in, V.A., Nikishin, E.M.: Questions on the convergence of multiple trigonometric series and spectral expansions. II. Usp. Mat. Nauk 32(1), 107–130 (1977). 10.4.8 zbMATHGoogle Scholar
 [B]Benedicks, M.: On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106, 180–183 (1985). 10.6.3 MathSciNetzbMATHCrossRefGoogle Scholar
 [C]Carleson, L.: On the convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966). 10.3.9 MathSciNetzbMATHCrossRefGoogle Scholar
 [Ca]Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables. Dover, New York (1995). 10.3.8 Google Scholar
 [CI]Chamizo, F., Iwaniec, H.: On the sphere problem. Rev. Mat. Iberoam. 11(2), 417–429 (1995). 10.6.6 MathSciNetzbMATHCrossRefGoogle Scholar
 [Ch]Chernoff, P.R.: Pointwise convergence of Fourier series. Am. Math. Mon. 87(5), 399–400 (1980). 10.3.4 MathSciNetzbMATHCrossRefGoogle Scholar
 [EF]Erdös, P., Fuchs, W.H.J.: On a problem of additive number theory. J. Lond. Math. Soc. 31, 67–73 (1956). 10.2.1, 10.6.6, 10.6 (Ex. 9) zbMATHCrossRefGoogle Scholar
 [HR]Hardy, G.H., Rogosinski, W.W.: Fourier Series. Cambridge University Press, Cambridge (1950). 10.3 (Ex. 5) zbMATHGoogle Scholar
 [LK]Hua, L.K.: Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie. Teubner Verlagsgesellschaft, Leipzig (1959). 10.6.6 zbMATHGoogle Scholar
 [Ke]Kendall, D.G.: On the number of lattice points inside a random oval. Q. J. Math. Oxf. Ser. 19, 1–26 (1948). 10.6.7 CrossRefGoogle Scholar
 [Luk]Lukacs, E.: Characteristic Functions. Hafner, New York (1970). 10.5.4 zbMATHGoogle Scholar
 [Na]Nazarov, F.L.: The Bang solution of the coefficient problem. St. Petersburg Math. J. 9(2), 407–419 (1998). 10.1.8 MathSciNetGoogle Scholar
 [PS]Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vols. I, II. Springer, Berlin (1998). 10.3.5 Google Scholar
 [Zy]Zygmund, A.: Trigonometric Series, vols. I, II. Cambridge University Press, New York (1959). 10.4.7 zbMATHGoogle Scholar