Fourier Series and the Fourier Transform

10.1.1

The norm in the space Open image in new window has an important specific feature: just like a norm in a finite dimensional Euclidean space, it is generated by a scalar product. The scalar product of functions f and g belonging to the (in general, complex) space Open image in new window is defined by the formula

$$\langle f,g\rangle=\int_Xf\overline{g}\,d\mu $$

(the product \(f\,\overline{g}\) is summable since \(2|f\,\overline{g}|\leqslant|f|^{2}+|g|^{2}\)).

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 left-hand side of the equation exists since the series converges and the scalar product is continuous).

We point out one more property of the norm in Open image in new window , the so-called parallelogram identity

Open image in new window

which is connected with the fact that the norm is generated by a scalar product.

The reader can easily verify that if a measure is non-degenerate (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 finite-dimensional 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.

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.

Theorem

Proof

10.1.3

Let \(\{e_{n}\}_{n\in\mathbb{N}}\) be an OS in the space Open image in new window . Obviously, there are functions in Open image in new window that cannot be represented as linear combinations of functions e _n . Therefore, the question naturally arises, what are the conditions under which a function Open image in new window is the sum of a series of the form \(\sum_{n=1}^{\infty} a_{n}e_{n}\). From the theorem proved above it follows that such a series can converge to f only if it coincides with the series \(\sum_{n=1}^{\infty} c_{n}(f)e_{n}\) whose coefficients are calculated by formula (2). Indeed, Eq. (3) shows that if a _m ≠c _m (f) and n⩾m, then

$$\Bigg\|f-\sum_{k=1}^na_ke_k \Bigg\|\geqslant\big|a_m-c_m(f)\big|\|e_m \|^2>0, $$

and, therefore, the series \(\sum_{n=1}^{\infty} a_{k}e_{k}\) cannot converge to f.

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.

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

Let S _n and T _n be the partial sums of series (5) and (5′), respectively. Then, for all \(n,p\in\mathbb{N}\), we have

$$\|S_{n+p}-S_n\|^2= \Bigg\|\sum_{k=n+1}^{n+p}a_ke_k\Bigg\|^2= \sum_{k=n+1}^{n+p}|a_k|^2 \|e_k\|^2=T_{n+p}-T_n. $$

It follows that the partial sums of series (5) and (5′) are fundamental simultaneously. Since the space Open image in new window is complete (see Theorem 9.1.3), we obtain the first assertion of the lemma. The concluding assertion follows from the fact that scalar multiplication of a convergent series by a function can be performed termwise, i.e., if S is the sum of series (5), then the relation

$$\langle S,e_m\rangle=\sum_{n=1}^{\infty} a_n\langle e_n,e_m\rangle= a_m \|e_m\|^2 $$

is valid for every \(m\in\mathbb{N}\). Thus, a _m =c _m (S) for all m, i.e., series (5) is the Fourier series of its sum. □

Theorem

(Riesz–Fischer⁴)

For every orthogonal system \(\{e_{n}\}_{n\in\mathbb{N}}\), the Fourier series of a function Open image in new window converges in norm and

$$ f=\sum_{n=1}^{\infty} c_n(f)e_n+h, \quad\text{where} \ h\perp e_n \ \text{for all}\ n\in\mathbb{N}. $$

(6)

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.

Definition

A family of functions {f _α } _α∈A in Open image in new window is called complete if the condition

Open image in new window

implies that f=0 almost everywhere, i.e., ∥f∥=0.

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)

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. □

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 non-zero 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 well-defined, 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

10.1.7

Let \(\{e_{k}\}_{k\in\mathbb{N}}\) and \(\{g_{n}\} _{n\in\mathbb{N}}\) be orthogonal systems in the spaces Open image in new window and  Open image in new window , respectively. We use these systems to construct an OS \(\{h_{k,n}\}_{k,n\in\mathbb{N}}\) in the space Open image in new window by putting

$$h_{k,n}(x,y)=e_k(x)g_n(y)\quad(x\in X, y\in Y). $$

Using Fubini’s theorem, we can easily verify that the functions h _k,n are square-summable and pairwise orthogonal. We will prove that the above construction preserves completeness.

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

Let f⊥h _k,n for all \(k,n\in\mathbb{N}\). This means that

$$\begin{aligned} &\int_{X\times Y}f(x,y)\overline{e_k(x)}\, \overline{g_n(y)}\,d(\mu\times \nu) (x,y) \\ &\quad = \int_X \biggl(\int_Yf(x,y)\overline{g_n(y)}\,d\nu(y) \biggr)\overline{e_k(x)}\, d\mu(x)=0 \end{aligned}$$

(7)

for all \(k,n\in\mathbb{N}\). We fix an arbitrary n and consider the function

$$x\mapsto \varphi _n(x)=\int_Yf(x,y) \overline{g_n(y)}\,d\nu(y). $$

This function is measurable by Corollary 2 to Tonelli’s theorem. Moreover, Open image in new window since

$$\big|\varphi _n(x)\big|\leqslant \biggl(\int_Y\big|f(x,y)\big|^2 \,d\nu(y) \biggr)^{1/2}\|g_n\|, $$

and, therefore,

$$\int_X\big|\varphi _n(x)\big|^2\,d\mu(x) \leqslant\int_X \biggl(\int_Y\big|f(x,y)\big|^2 \,d\nu (y) \biggr)\, d\mu(x)\|g_n\|^2<+\infty. $$

Equation (7) means that the Fourier coefficients of φ _n with respect to the system \(\{e_{k}\}_{k\in\mathbb{N}}\) are zero. Since the system is complete, we have φ _n (x)=0 almost everywhere. Since this is true for all indices n, we have

$$ \sum_{n=1}^{\infty}\big|\varphi _n(x)\big|^2=0 \quad\text{almost everywhere on}\ X. $$

(8)

Since ∫ _X ∫ _Y |f(x,y)|² dν(y) dμ(x)<+∞, Fubini’s theorem implies that ∫ _Y |f(x,y)|² dν(y)<+∞ almost everywhere. In other words, the function y↦f _x (y)=f(x,y) is square-summable for almost all x. The numbers φ _n (x) are simply the Fourier coefficients of this function with respect to the system \(\{g_{n}\}_{n\in\mathbb{N}}\). Since the system \(\{g_{n}\}_{n\in \mathbb{N}}\) is complete, Eq. (8) means that

$$\int_Y\big|f(x,y)\big|^2\,d\nu(y)=\|f_x \|^2=\sum_{n=1}^{\infty}\big| \varphi _n(x)\big|^2=0\quad \text{almost everywhere on}\ X. $$

Integrating the above equation over X, we obtain

$$0=\int_X\int_Y\big|f(x,y)\big|^2 \,d\nu(y)\,d\mu(x)=\|f\|^2. $$

Consequently, f=0 almost everywhere, which proves that the system \(\{ h_{k,n}\}_{k,n\in\mathbb{N}} \) is complete. □

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 square-summable 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 square-summable 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. Nazarov⁶ [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).

Proof

We consider only the real case, leaving the complex case to the reader (see Exercises 6 and 7).

Now, we consider the function Φ of class \(C^{2}(\mathbb{R})\) such that |Φ′|,|Φ″|⩽1 (the choice of Φ will be specified later). Since |Φ(u)|⩽|Φ(0)|+|u| and the measure is finite, the integral I(f)=∫ _X Φ(f) dμ is finite for every function Open image in new window . Obviously, the integral continuously depends on f and so, by the Weierstrass extreme value theorem, it assumes its maximum value on A: there exists a sequence of signs ε={ε _n } such that I(f _ε )⩾I(f) for every function f in A. We show that the required function has the form F _a =Φ′(f _ε ) for an appropriate choice of Φ. Since |F _a |⩽sup|Φ′|⩽1, it only remains for us to estimate the Fourier coefficients c _n (F _a ). To this end, we use the fact that the replacement of ε _n by −ε _n leaves a function in the class A and, therefore, does not increase the integral I,

$$\int_X \bigl(\Phi (f_{\varepsilon } )-\Phi (f_{\varepsilon }-2\varepsilon _na_ne_n ) \bigr)\, d \mu\geqslant0. $$

The application of the Taylor formula to the integrand leads to the inequality

$$ \int_X \biggl(2\varepsilon _na_n e_n\Phi'(f_{\varepsilon })- \frac{1}{2}(2 \varepsilon _na_ne_n)^2 \Phi''(g_n) \biggr)d\mu\geqslant0, $$

(9)

where g _n is a function whose values lie between f _ε and f _ε −2ε _n a _n e _n .

EXERCISES

10.2.1

Trigonometric Systems. The most important orthogonal systems are the following real and complex trigonometric systems in the space Open image in new window :

$$1,\quad \cos\frac{\pi x}{\ell},\quad \sin\frac{\pi x}{\ell},\quad \ldots, \quad \cos \frac{\pi nx}{\ell},\quad \sin\frac{\pi nx}{\ell},\quad \ldots \quad\text{and} \quad \bigl\{e^{\frac{\pi inx}{\ell}}\bigr\}_{n\in\mathbb{Z}}. $$

We leave the simple verification of the orthogonality to the reader. The Fourier series with respect to these systems have, respectively, the form

$$A(f)+\sum_{n=1}^{\infty} \biggl(a_n(f)\cos\frac{\pi nx}{\ell}+ b_n(f)\sin \frac{\pi nx}{\ell}\biggr)\quad\text{and}\quad \sum _{n=-\infty}^{\infty} c_n(f) e^{\frac{\pi inx}{\ell}}, $$

where the Fourier coefficients are calculated by the formulas

$$\begin{aligned} A(f)=&\frac{1}{2\ell}\int_a^{a+2\ell}f(x)\,dx, \\a_n(f)=&\frac{1}{\ell}\int_a^{a+2\ell}f(x) \cos\frac{\pi nx}{\ell}\, dx,\\b_n(f)=&\frac{1}{\ell}\int _a^{a+2\ell}f(x)\sin\frac{\pi nx}{\ell}\,dx \quad(n\in\mathbb{N});\\c_n(f)=&\frac{1}{2\ell}\int_a^{a+2\ell}f(x) e^{-\frac{\pi inx}{\ell}}\,dx\quad(n\in\mathbb{Z}). \end{aligned}$$

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).

The transition from the expansion in one system to the expansion in a different system proceeds as follows. For a function Open image in new window , we define a function g by putting \(g(y)=f(\frac{\ell}{\pi}y)\), where y∈(0,2π). It is clear that Open image in new window . There is an obvious relation connecting the Fourier coefficients of these functions (with respect to the corresponding systems):

$$c_k(f)=\frac{1}{2\ell}\int_0^{2\ell}f(x) e^{-\frac{\pi ikx}{\ell}}\,dx= \frac{1}{2\pi}\int_0^{2\pi}f \biggl(\frac{\ell}{\pi}y \biggr) e^{-iky}\,dy= \widehat{g}(k) $$

for each \(k\in\mathbb{Z}\). Consequently,

$$\sum_{|k|\leqslant n}c_k(f) e^{\frac{\pi ikx}{\ell}}= \sum _{|k|\leqslant n}\widehat{g}(k) e^{\frac{\pi ikx}{\ell}}= \sum _{|k|\leqslant n}\widehat{g}(k) e^{iky}, $$

i.e., the partial sums of the Fourier series of the functions f and g at the corresponding points coincide. From this, it follows, in particular, that both series converge simultaneously and their sums coincide (or do not coincide) simultaneously with the values of the functions f and g. Thus, the transition from f to g makes it possible to reduce the study of a Fourier series in a system with an arbitrary period to the study of a Fourier series in the 2π-periodic system.

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. □

Example 1

Example 2

As we have seen (see Corollary 9.2.4), 2π-periodic 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.

We will assume that a function Open image in new window is extended by periodicity from [−π,π] to \(\mathbb{R}\). Let \(h\in\mathbb{R}\), and let f _h be the corresponding translation of f, i.e., f _h (x)=f(x−h) for \(x\in\mathbb{R}\). It can easily be verified that \(\widehat {f_{h}}(k)=e^{-ikh}\widehat{f}(k)\). Therefore, by Parseval’s identity, we obtain

$$\|f_h-f\|^2=2\pi\sum_{k=-\infty}^{\infty}\big| \widehat{f}(k)\big|^2 \big|e^{-ikh}-1 \big|^2= 8\pi\sum _{k=-\infty}^{\infty}\big|\widehat{f}(k)\big|^2 \sin^2\frac{kh}{2}. $$

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.

Let the Fourier coefficients of a function φ in Open image in new window be non-negative. Then the inequality

$$\frac{1}{2\pi}\int_{-\pi}^{\pi}\big| \varphi (t)\big|^2\,dt\leqslant \frac{3}{2\alpha }\int_{-\alpha }^{\alpha }\big| \varphi (t)\big|^2\,dt $$

is valid for every α∈(0,π).

Since the function \(h(t)= (1-\frac{|t|}{\alpha })_{+}\) does not exceed 1, it is sufficient for us to estimate the integral \(I=\int_{-\pi}^{\pi}|\varphi (t)h(t)|^{2}\,dt\) from below. The product f=φh, obviously, belongs to Open image in new window . We calculate its Fourier coefficients (in what follows, e _n (t)=e ^int ),

$$\begin{aligned} \widehat{f}(n)=&\frac{1}{2\pi}\langle \varphi h,e_n\rangle= \frac{1}{2\pi}\langle \varphi ,he_n\rangle=\frac{1}{2\pi} \sum _{k=-\infty}^{\infty}\widehat{\varphi }(k)\langle e_k,he_n\rangle\\=& \sum_{k=-\infty}^{\infty} \widehat{\varphi }(k)\,\widehat{h}(n-k) =\sum_{k+j=n}\widehat{\varphi }(k)\,\widehat{h}(j). \end{aligned}$$

Now, by Parseval’s identity we obtain

$$I=2\pi\sum_{n=-\infty}^{\infty}\big| \widehat{f}(n)\big|^2=2\pi\sum_{n=-\infty}^{\infty} \bigg|\sum_{k+j=n}\widehat{\varphi }(k)\,\widehat{h}(j)\bigg|^2. $$

A direct calculation shows that \(\widehat{h}(j)\geqslant0\) for all \(j\in\mathbb{Z}\) (this also follows from the result of Example 2 of Sect. 4.6.6 since the function h is convex on (0,π)). Therefore, replacing the square of the sum by the sum of squares (here we use the inequalities \(\widehat{\varphi }(k)\geqslant0\)), we obtain

$$\begin{aligned} I\geqslant&2\pi\sum_{n=-\infty}^{\infty} \sum _{k+j=n}\widehat{\varphi }^2(k)\, \widehat{h}^2(j)= 2\pi\sum_{k=-\infty}^{\infty} \widehat{\varphi }^2(k)\sum_{j=-\infty}^{\infty} \widehat{h}^2(j) \\=&\frac{1}{2\pi}\int_{-\pi}^{\pi}\big| \varphi (t)\big|^2\,dt\int_{-\pi}^{\pi} h^2(t)\,dt= \frac{\alpha }{3\pi}\int_{-\pi}^{\pi}\big| \varphi (t)\big|^2\,dt. \end{aligned}$$

Thus,

$$\int_{-\alpha }^{\alpha }\big|\varphi (t)\big|^2dt\geqslant I \geqslant \frac{\alpha }{3\pi}\int_{-\pi}^{\pi}\big| \varphi (t)\big|^2\,dt. $$

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

Considering the product of m copies of the complex trigonometric system (see Sect. 10.1.7), we obtain its multi-dimensional analog in the space Open image in new window , where Q=(−π,π) ^m (a multi-dimensional version of the real trigonometric system is quite cumbersome and we do not consider it). The new system consists of the complex exponential functions e _n numbered by multi-indices n=(n ₁,…,n _m ):

$$e_n(x)=e^{i\langle n,x\rangle},\quad \text{where}\ x\in Q,\ n\in \mathbb{Z}^m. $$

The Fourier coefficients of a function Open image in new window in this system are calculated by the formulas

$$\widehat{f}(n)=\frac{\langle f,e_n\rangle}{\|e_n\|^2}=\frac{1}{(2\pi )^m}\int_Qf(x) e^{-i\langle n,x\rangle}\,dx \quad\bigl(n\in\mathbb{Z}^m\bigr). $$

From Theorem 10.1.7, it follows that the system \(\{e^{i\langle n,x\rangle }\}_{n\in\mathbb{Z}^{m}}\) is complete, which implies Parseval’s identity

Open image in new window

Of course, the cube Q=(−π,π) ^m in the two last formulas can be replaced by a shifted cube.

Example

Let 0<ρ⩽π. We consider the function Open image in new window that is equal to 1/∥x∥ for ∥x∥<ρ and vanishes on (−π,π)³∖B(0,ρ). Its norm is easily calculated in spherical coordinates,

$$\|f\|^2=\int_{B(0,\rho)}\frac{1}{\|x\|^2}\,dx= 4\pi \int_0^{\rho}\frac{1}{r^2}\,r^2 \,dr=4\pi\rho. $$

To calculate the Fourier coefficients, we use the formula obtained in the example of Sect. 6.2.5 with f ₀(r)=1/r on (0,ρ), f ₀(r)=0 for r⩾ρ and y=n/2π:

$$\begin{aligned} \widehat{f}(n)=&\frac{1}{(2\pi)^3}\int_{B(0,\rho)} \frac{1}{\|x\|} e^{-i\langle n,x\rangle}\,dx=\frac{1}{2\pi^2\|n\|} \int _0^{\rho}\frac{1}{r}r\sin\bigl(\|n\|r\bigr)\,dr\\=& \biggl(\frac{\sin\frac{\rho}{2}\|n\|}{\pi\|n\|} \biggr)^2 \end{aligned}$$

if n≠0 and \(\widehat{f}(0)=\rho^{2}/4\pi^{2}\). By Parseval’s identity for the function f, we obtain

$$4\pi\rho=(2\pi)^3\sum_{n\in\mathbb{Z}^3} \biggl( \frac{\sin\frac{\rho}{2}\|n\|}{\pi\|n\|} \biggr)^4. $$

Thus, the identity

$$\frac{\pi^2}{t^3}=\sum_{n\in\mathbb{Z}^3} \biggl( \frac{\sin\|n\|t}{\|n\| t} \biggr)^4 $$

is valid for \(t=\frac{\rho}{2}\in(0,\frac{\pi}{2}]\) (the summand for n=0 is equal to 1).

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 non-trivial 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 non-trivial 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 non-trivial invariant sets (such maps are called ergodic). Let us prove this.

10.2.4

We will now give other examples of orthogonal systems. Let P _n (x)=((x ²−1) ⁿ )⁽ⁿ⁾, n=0,1,…. The polynomials P _n are called the Legendre polynomials. Obviously, deg P _n =n, and so every polynomial is a linear combination of Legendre polynomials, which form an orthogonal system in the space Open image in new window . Indeed, for m<n, we have

$$\begin{aligned} \langle P_m,P_n\rangle=&\int_{-1}^1P_m(x) \bigl(\bigl(x^2-1\bigr)^n \bigr)^{(n)}\,dx\\=& P_m(x) \bigl(\bigl(x^2-1\bigr)^n \bigr)^{(n-1)}\bigg\vert _{-1}^1 -\int_{-1}^1P'_m(x) \bigl(\bigl(x^2-1\bigr)^n \bigr)^{(n-1)}\,dx\\=&-\int_{-1}^1P'_m(x) \bigl( \bigl(x^2-1\bigr)^n \bigr)^{(n-1)}\,dx. \end{aligned}$$

Integrating by parts n times, we arrive at the equation

$$\langle P_m,P_n\rangle=(-1)^n\int_{-1}^1P^{(n)}_m(x) \bigl(x^2-1\bigr)^n\,dx, $$

where \(P^{(n)}_{m}(x)\equiv0\), since deg P _m <n. Thus, 〈P _m ,P _n 〉=0 for m≠n.

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. □

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 non-empty 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 one-element 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 \(\sum_{s=j}^{k-1}\chi_{\Delta_{n,s}}\) 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’s¹⁰ inequality)

Let f ₁,…,f _n in Open image in new window be independent and have zero means, ∫ _X f ₁ dμ=⋯=∫ _X f _n  dμ=0. Then the inequality

$$\mu \Bigl( \Bigl\{x\in X\, |\,\max_{1\leqslant k\leqslant n} \big|f_1(x)+ \cdots+f_k(x)\big| \geqslant t \Bigr\} \Bigr)\leqslant \frac{1}{t^2} \sum_{k=1}^n\int_X f_k^2\,d\mu $$

holds for every t>0.

Proof

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. □

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

10.3.1

We recall that, according to the general definition 10.2.1, the Fourier series of a function Open image in new window in the systems

$$1,\quad \cos x,\quad \sin x,\quad \ldots,\quad \cos nx,\quad \sin nx,\quad \ldots,\quad\text{and} \quad \bigl\{e^{inx}\bigr\}_{n\in\mathbb{Z}} $$

have, respectively, the forms

$$ A(f)+\sum_{n=1}^{\infty}\bigl(a_n(f) \cos nx+b_n(f)\sin nx\bigr) $$

(1)

and

$$ \sum_{n=-\infty}^{\infty}\widehat{f}(n) e^{inx}, $$

(1′)

where the Fourier coefficients are calculated by the formulas

$$\begin{aligned} A(f)=&\frac{1}{2\pi}\int_0^{2\pi}f(x)\,dx, \\a_n(f)=&\frac{1}{\pi}\int_0^{2\pi}f(x) \cos nx\,dx, \end{aligned}$$

(2)

$$\begin{aligned} b_n(f)=&\frac{1}{\pi}\int_0^{2\pi}f(x) \sin nx\,dx \quad(n\in\mathbb{N}); \\\widehat{f}(n)=&\frac{1}{2\pi}\int_0^{2\pi}f(x) e^{-inx}dx\quad(n\in \mathbb{Z}). \end{aligned}$$

(2′)

Unlike the previous section, where we considered only functions of class Open image in new window , here we will deal with arbitrary functions summable on (0,2π). It is obvious that, in this case, the integrands in formulas (2) and (2′) will also be summable. Therefore, we keep the terminology introduced above (a Fourier coefficient, a Fourier series) for the functions in Open image in new window . We are now interested not in convergence in the Open image in new window -norm, but in other types of convergence, and first of all, pointwise convergence. Here, by the sum of the series (1′), we always mean the limit of the symmetric partial sums

$$ S_n(f,x)=\sum_{|k|\leqslant n}\widehat{f}(k) e^{ikx}, $$

(3)

which are also called the Fourier sums of the function f. As noted in Sect. 10.2.1, the partial sums of series (1) and (1′) are equal. Thus, all results obtained for one of the series are valid for the other one. In the sequel, we will mainly consider series (1′) because this leads to some technical simplifications.

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 2π-periodic 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 2π-periodicity). 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).

The properties connecting Fourier coefficients with translation, differentiation, and convolution play an important role. We recall that the translation f _h of a function Open image in new window corresponding to a number h is defined by the formula f _h (x)=f(x−h). Making the change of variable x−h↦x in the integral \(\int_{0}^{2\pi}f(x-h) e^{-inx}\,dx\), we arrive at the formula

(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

$$\widehat{f'}(n)=in\widehat{f}(n)\quad(n\in\mathbb{Z}) $$

(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.

(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

$$\widehat{f*g}(n)=2\pi\widehat{f}(n)\cdot\widehat{g}(n)\quad\text{for all}\ n \in\mathbb{Z} $$

(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,

$$\begin{aligned} \widehat{f*g}(n)=&\frac{1}{2\pi}\int_{-\pi}^{\pi}(f*g) (x) e^{-inx}\,dx\\=& \frac{1}{2\pi}\int_{-\pi}^{\pi} \biggl(\int_{-\pi}^{\pi} f(x-t)g(t)\, dt \biggr)e^{-inx}\,dx \\=&\frac{1}{2\pi}\int_{-\pi}^{\pi} g(t) e^{-int} \biggl(\int_{-\pi}^{\pi} f(x-t) e^{-in(x-t)}dx \biggr)dt \\=&\frac{1}{2\pi}\int_{-\pi}^{\pi} g(t) e^{-int} \biggl(\int_{-\pi}^{\pi} f(u) e^{-inu}du \biggr)dt=2\pi\widehat {g}(n)\cdot\widehat{f} (n). \end{aligned}$$

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 square-summable 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.

We begin the investigation of a Fourier series’ convergence with the derivation of an important formula for its partial sums discovered by Dirichlet. Relying on formula (2′), we transform Eq. (3) as follows:

$$S_n(f,x)=\sum_{|k|\leqslant n} \biggl( \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-ikt}\,dt \biggr) e^{ikx}=\frac{1}{2\pi}\int _{-\pi}^{\pi} f(t)\sum_{|k|\leqslant n}e^{ik(x-t)} \,dt. $$

The function

$$ D_n(u)=\frac{1}{2\pi}\sum_{|k|\leqslant n}e^{iku}= \frac{1}{2\pi}+ \frac{1}{\pi}\sum_{k=1}^n \cos ku $$

(4)

is called the nth Dirichlet kernel. Obviously, the Dirichlet kernel is even and periodic. Summing the geometric sequence ∑_|k|⩽n e ^iku , we obtain

$$ D_n(u)=\frac{\sin (n+\frac{1}{2} )u}{2\pi\sin\frac{u}{2}}\quad \text{for }\ u\notin2\pi\mathbb{Z}. $$

(4′)

From this, we see that the function D _n is strongly oscillating for large n, and, in a neighborhood of zero, it takes extreme values with alternating signs and absolute values comparable with \(\max D_{n}=D_{n}(0)=\frac{1}{\pi} (n+\frac{1}{2} )\) (see Fig. 10.1).

Open image in new window

However, D _n does not satisfy the most important property of an approximate identity, namely, the positivity. Moreover, the Dirichlet kernels do not satisfy the periodic analog of condition (a′) of Sect. 7.6.1, i.e., they have unbounded Open image in new window -norms. Indeed,

$$\begin{aligned} \int_{-\pi}^{\pi}\big|D_n(u)\big|\,du=& \int _0^{\pi}\frac{|\!\sin(n+\frac{1}{2})u|}{\pi\sin\frac{u}{2}}\,du\geqslant \frac{2}{\pi}\int_0^{\pi}\frac{|\!\sin(n+\frac{1}{2})u|}{u} \,du \\=&\frac{2}{\pi}\int_0^{\pi(n+\frac{1}{2})} \frac{|\!\sin v|}{v}\,dv\geqslant \frac{2}{\pi}\sum _{k=1}^n\int_{(k-1)\pi}^{k\pi} \frac{|\!\sin v|}{k\pi}\,dv =\frac{4}{\pi^2}\sum_{k=1}^n \frac{1}{k}. \end{aligned}$$

Since \(\sum_{k=1}^{n}\frac{1}{k}\geqslant\int_{1}^{n}\frac{1}{x}\,dx=\ln n\), we have \(\|D_{n}\|_{1}\geqslant\frac{4}{\pi^{2}}\ln n\) (see also Exercise 11).

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 Dini¹² test and the Dirichlet–Jordan¹³ 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)

If a function Open image in new window satisfies the Dini condition

$$\int_0^{\pi}\biggl \vert \frac{f(x+u)+f(x-u)}{2}-C \biggr \vert \,\frac{du}{u}<+\infty $$

at a point \(x\in\mathbb{R}\) for some \(C\in\mathbb{C}\), then its Fourier series converges to C at the point x.

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, \(|S_{n}(f,x)|\leqslant\sup_{\mathbb{R}}|f|+2\mathbf{V}_{-\pi}^{\pi}(f)\).

Proof

10.3.5

We give some examples of Fourier series expansions.

Example 1

We supplement Example 1 of Sect. 10.2.1 as follows: since the periodic function equal x on (−π,π) is differentiable at all points distinct from (2k+1)π \((k\in\mathbb{Z})\), its Fourier series converges not only in the Open image in new window -norm, but also pointwise,

$$x=2\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n} \sin nx\quad \text{for}\ x\in(-\pi,\pi). $$

At the points (2k+1)π, the sum of the series is equal to the average of the one-sided limits of the function. At \(x=\frac{\pi}{2}\), the Fourier series expansion yields the relations

$$\frac{\pi}{4}=\sum_{m=0}^{\infty}(-1)^m \frac{1}{2m+1}. $$

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

Let Open image in new window . Then the equation

$$\int_a^b f(x)\,dx=\sum _{n=-\infty}^{\infty}\widehat{f}(n)\int_a^b e^{inx}\,dx $$

(where the sum is regarded as the limit of the symmetric partial sums) is valid for all \(a,b\in \mathbb{R}\).

Proof

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.

Proof

Theorem 2

Let Open image in new window , and let g be a bounded (measurable and periodic) function whose Fourier sums S _n (g,x) are uniformly bounded (with respect to x and n). Then the following Parseval identity is valid:

$$\int_{-\pi}^{\pi} f(x)\overline{g}(x)\,dx=2\pi\sum _{n=-\infty}^{\infty} \widehat{f}(n)\overline{ \widehat{g}(n)}. $$

Proof

Since Open image in new window , the sums S _n (g) converge to g in the Open image in new window -norm and, a fortiori, in measure. This implies, as one can easily verify, that

$$f(x)\overline{S_n}(g,x)\to f(x)\overline{g}(x)\quad \text{in measure.} $$

Therefore, we can use Lebesgue’s theorem and pass to the limit on the right-hand side of the equation

$$\int_{-\pi}^{\pi} f(x)\overline{S_n}(g,x) \,dx= 2\pi\sum_{|k|\leqslant n} \widehat{f}(k)\overline{\widehat{g}(k)}, $$

as required. □

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

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 \(\widehat{f}(n)=O(|n|^{-r-2})\) 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 r-fold 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 \(r\in\mathbb{N}\).

Theorem 2

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

As we have already mentioned, the Fourier series of a periodic continuous function may diverge (compare this with the result of Exercise 5). There are several such examples. We give here a slight modification of an example suggested by Schwartz. We define an even function \(f\in\widetilde{C}\) whose oscillation frequency increases rapidly when approaching zero. More precisely, we will assume that

$$f(0)=0 \quad \text{and}\quad f(t)=\frac{1}{\sqrt{k}}\sin n_kt \quad \text{for}\ t\in[t_k,t_{k-1}],\ k=2,3,\ldots, $$

where n _k =2 ^k!, t _k =2π/n _k for \(k\in\mathbb{N}\) (see Fig. 10.2).

Open image in new window

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

(Denjoy¹⁵–Luzin)

Proof

EXERCISES

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 Cesaro¹⁶ is, by definition, the limit \(\lim_{n\to\infty}\frac{1}{n}(a_{0}+\cdots+a_{n-1})\). 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 \(\frac{1}{n}(a_{0}+\cdots+a_{n-1})\underset{n\to\infty}{\longrightarrow} a\) (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)

Let Open image in new window and \(x\in\mathbb{R} \). Then:

1. (a)

   if the limits L _±=lim _t→x±0 f(t) exist and are finite, then \(\sigma _{n}(f,x)\underset{n\to\infty}{\longrightarrow}\frac{L_{+}+L_{-}}{2}\);

    
2. (b)

   if \(f\in\widetilde{C}\), then \(\sigma _{n}(f)\underset{n\to\infty}{\rightrightarrows} f\) on \(\mathbb{R}\);

    
3. (c)

   if Open image in new window for some p∈[1,+∞), then \(\|\sigma _{n}(f)-f\|_{p}\underset{n\to\infty}{\longrightarrow}0\);

    
4. (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 one-sided 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.

Remark

The third and, obviously, the fourth statements of the theorem give new proofs of the uniqueness theorem. Indeed, if functions Open image in new window have the same Fourier coefficients, then σ _n (f)=σ _n (g) for all n. Therefore, statement (c) of the theorem for p=1 implies the relation

$$\|f-g\|_1=\lim_{n\to\infty}\big\|\sigma _n(f)- \sigma _n(g)\big\|_1=0. $$

Consequently, the functions f and g are equal almost everywhere.

Corollary 1

The Fourier series of an absolutely continuous periodic function f converges to f uniformly on  \(\mathbb{R}\).

Proof

By the second statement of the Fejér theorem, it is sufficient to verify that the difference S _n (f)−σ _n (f) converges uniformly to zero. It is clear that

$$\big|S_n(f,x)-\sigma _n(f,x) \big|= \Bigg|\sum _{k=-n}^n\frac{|k|}{n}\widehat{f}(k) e^{ikx} \Bigg|\leqslant\frac{1}{n}\sum_{k=-n}^n|k|\big| \widehat{f}(k)\big|. $$

By assumption, we have \(f(x)=f(0)+\int_{0}^{x} g(t)\,dt\), where Open image in new window , and, therefore (see property (d) of Sect. 10.3.2) \(|k\widehat {f}(k)|=|\widehat{f}'(k)|\). It remains to use the permanence of the method of arithmetic means: since \(\widehat{f}'(k)\underset{|k|\to+\infty}{\longrightarrow}0\), we have

$$\max_{x\in\mathbb{R}} \big|S_n(f,x)-\sigma _n(f,x) \big| \leqslant\frac {1}{n}\sum_{k=-n}^n\big| \widehat{f'}(k)\big| \underset{n\to\infty}{\longrightarrow}0. $$

 □

Corollary 2

Proof

10.4.2

The non-negativity of the Fejér kernels yields interesting properties of the Fourier series. For example, if the coefficients of the Fourier series of a function are non-negative, then the Fourier series converges absolutely if the function is bounded in a neighborhood of zero. Indeed, let a function Open image in new window be such that \(\widehat{f}(k)\geqslant0\) for all \(k\in\mathbb{Z}\) and |f(x)|⩽C if |x|<δ. Since 0⩽Φ _n (x)⩽C _δ for δ⩽|x|⩽π, we obtain

$$\begin{aligned} \big|\sigma _n(f,0)\big|=& \bigg|\int_{-\pi}^{\pi} f(x) \Phi_n(x)\,dx \bigg|\leqslant \int_{-\delta }^{\delta } C\Phi_n(x)\,dx+\int_{\delta \leqslant|x|\leqslant\pi}C_{\delta }\big|f(x)\big| \, dx\\\leqslant& C+C_{\delta }\|f\|_ 1. \end{aligned}$$

Therefore,

$$\sum_{|k|<\frac{n}{2}}\widehat{f}(k)\leqslant2\sum _{|k|<n} \biggl(1-\frac {|k|}{n} \biggr) \widehat{f}(k) =2 \sigma _n(f,0)\leqslant2\bigl(C+C_{\delta }\|f\|_1\bigr). $$

Since this inequality is fulfilled for all n, the series \(\sum_{k=-\infty}^{\infty}\widehat{f}(k)\) converges.

Before passing to the study of the sum of series (4), we prove a lemma on numerical series.

Lemma

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 non-negative and summable on (−π,π) and series (4) is the Fourier series of f.

Proof

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 well-known 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.

Turning to Fourier series, we see that, to each function Open image in new window , we must assign the sums

$$S_{M,\varepsilon }(f,x)=\sum_{n=-\infty}^{\infty} M\bigl( \varepsilon |n|\bigr)\widehat{f}(n) e^{inx}\quad (x\in\mathbb{R}). $$

To find integral representations for them, we use property (e) of Sect. 10.3.2, \(\widehat{f*g}(n)=2\pi\widehat{f}(n)\cdot\widehat{g}(n)\), which allows us to regard S _M,ε (f) as the sum of Fourier series of the convolution f∗ω _ε , where

$$\omega_{\varepsilon }(x)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty} M\bigl(\varepsilon |n|\bigr) e^{inx}= \frac{1}{2\pi}+\frac{1}{\pi}\sum _{n=1}^{\infty} M(\varepsilon n)\cos nx \quad (x\in\mathbb{R}). $$

By the uniqueness theorem (Corollary 1 of Sect. 10.3.6), the functions f∗ω _ε and S _M,ε (f) coincide almost everywhere. Since the functions are continuous, they coincide everywhere.

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 \(\int_{-\pi}^{\pi}\Omega_{\varepsilon }(x)\,dx\leqslant2\).

Proof

Theorem

Let Open image in new window , \(x\in\mathbb{R}\), and let the function M be summable on [0,+∞) and satisfy the conditions of the lemma. Then:

1. (a)

   if the limits L _±=lim _t→x±0 f(t) exist and are finite, then \(S_{M,\varepsilon }(f,x)\underset{\varepsilon \to0}{\longrightarrow}\frac{L_{+}+L_{-}}{2}\);

    
2. (b)

   if \(f\in\widetilde{C}\), then \(S_{M,\varepsilon }(f)\underset{\varepsilon \to0}{\rightrightarrows} f\) on \(\mathbb{R}\);

    
3. (c)

   if Open image in new window for some p∈[1,+∞), then \(\|S_{M,\varepsilon }(f)-f\|_{p}\underset{\varepsilon \to0}{\longrightarrow}0\);

    
4. (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 hump-shaped 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 one-dimensional situation.

We introduce the necessary notation. When speaking of periodic functions in \(\mathbb{R}^{m}\), we will always mean 2π-periodicity with respect to each variable. By \(\widetilde{C}(\mathbb{R}^{m})\) and \(\widetilde{C}^{r}(\mathbb{R}^{m})\) (\(r\in\mathbb{N}\)), we denote the classes of periodic continuous and periodic smooth functions, respectively, defined on \(\mathbb{R}^{m}\); by Open image in new window (1⩽p⩽+∞), we denote the class of periodic pth power summable functions on the cube Q=(−π,π) ^m . For a function f in Open image in new window , we denote the Open image in new window -norm of its restriction to Q by ∥f∥ _p . For a function Open image in new window and a multi-index \(n\in\mathbb{Z}^{m}\),

$$\widehat{f}(n)=\frac{1}{(2\pi)^m}\int_Q f(x) e^{-i\langle n,x\rangle}\,dx $$

is the nth Fourier coefficient of f.

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 one-dimensional 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 one-dimensional 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.

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 |).

For a function Open image in new window and a multi-index \(n=(n_{1},\ldots,n_{m})\in\mathbb{Z}^{m}_{+}\), we consider the partial sum

$$S_n(f,x)=\sum_{|k|\leqslant n}\widehat{f}(k) e^{i\langle k,x\rangle} $$

of the Fourier series over the corresponding “rectangle”. Reasoning as in Sect. 10.3.3, we can easily verify that this sum is the convolution of f and a multi-dimensional Dirichlet kernel equal to the product to the product of one-dimensional Dirichlet kernels, S _n (f)=f∗D _n , where

$$D_n(u)=D_{n_1}(u_1)\cdots D_{n_m}(u_m) \quad\bigl(u=(u_1,\ldots,u_m)\bigr). $$

As in the one-dimensional case, the kernel D _n satisfies the equation (below, Q=(−π,π) ^m )

$$ \int_Q D_n(u)\,du=1. $$

(6)

It was noted in Sect. 10.3.3 that the Open image in new window -norms of one-dimensional Dirichlet kernels have logarithmic rate of growth. The same is also true for the Dirichlet kernels corresponding to rectangles,

$$\|D_n\|_1=\int_Q\big|D_n(u)\big| \,du=\prod_{j=1}^m\int _{-\pi}^{\pi} \big|D_{n_j}(u_j)\big| \,du_j \asymp\prod_{j=1}^m \ln n_j. $$

Hence it follows that the following estimate is valid for periodic bounded (in particular, continuous) functions and n ₁,…,n _m >1:

$$ \big\|S_n(f)\big\|_{\infty}\leqslant C_m \Biggl(\prod _{j=1}^m\ln n_j \Biggr)\|f\| _{\infty}. $$

(7)

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 one-dimensional 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. Fefferman¹⁸ 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

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

Corollary

Functions in Open image in new window coincide almost everywhere if they have the same Fourier coefficients.

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 two-dimensional 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

10.5.1

We introduce one of the main concepts of this chapter.

Definition

The Fourier transform \(\widehat{f}\) of a function f in Open image in new window is defined by the formula

$$\widehat{f}(y)=\int_{\mathbb{R}^m}f(x) e^{-2\pi i\langle y,x\rangle}\,dx \quad \bigl(y\in\mathbb{R}^m\bigr) $$

(here, as before, 〈y,x〉 is the scalar product of vectors y and x).

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. □

Example 1

The Fourier transform of the characteristic function χ of the interval (−1,1) is calculated very simply:

$$\widehat{\chi}(y)=\int_{-\infty}^{\infty}\chi(x) e^{-2\pi iyx}\,dx= \int_{-1}^1 e^{-2\pi i yx}\,dx=\frac{\sin2\pi y}{\pi y}. $$

We remark that Open image in new window (this is established in Example 1 of Sect. 4.6.6).

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.

Example 6

Let f be a function in Open image in new window equal to zero outside (−π,π), and let f ₀ be its 2π-extension from this interval to \(\mathbb{R}\) ( Open image in new window ). The Fourier coefficients of f ₀ can easily be expressed in terms of the Fourier transform of f, namely, \(\widehat{f_{0}}(n)=\frac{1}{2\pi}\widehat{f} (\frac{n}{2\pi} )\) for all \(n\in\mathbb{Z}\). We also consider the function g(x)=e ^itx for x∈(−π,π), where t is a fixed number. Since the Fourier coefficients of g are equal to

$$\widehat{g}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{i(t-n)x}\,dx= \frac{\sin\pi(t-n)}{\pi(t-n)}, $$

we obtain by Parseval’s generalized identity (see Theorem 2 of Sect. 10.3.6)

$$\begin{aligned} \widehat{f} \biggl(\frac{t}{2\pi} \biggr)=&\int_{-\pi}^{\pi} f_0(x)\overline{g}(x)\,dx= 2\pi\sum_{n=-\infty}^{\infty} \widehat{f_0}(n)\overline{\widehat{g}(n)}\\=& \sum _{n=-\infty}^{\infty}\widehat{f} \biggl(\frac{n}{2\pi} \biggr) \frac{\sin\pi(t-n)}{\pi(t-n)}. \end{aligned}$$

Thus, the following sampling formula is valid for the function \(F(t)=\widehat{f}(t/2\pi)\):

$$F(t)=\sum_{n=-\infty}^{\infty} F(n) \frac{\sin\pi(t-n)}{\pi(t-n)}= \frac{\sin\pi t}{\pi}\sum_{n=-\infty}^{\infty} \frac{(-1)^n}{t-n}F(n). $$

This formula allows one to recover the value of a function F at an arbitrary point t knowing the values of F on the integer lattice. This fact plays a fundamental role in optics and radio engineering because it is easier to deal with a discrete system of values than with a continuously varying signal. A multi-dimensional version of the sampling formula is given in Exercise 3.

10.5.2

We establish elementary relations between differentiation and the Fourier transform.

Theorem

Let Open image in new window . Then:

1. (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. (2)
   if the product ∥x∥f(x) is summable, then \(\widehat{f}\in C^{1}(\mathbb{R}^{m})\) and the equation

   $$\frac{\partial\widehat{f}(y)}{\partial y_k}=-2\pi i\widehat {f}_k(y),\quad {\textit{where}}\ f_k(x)=x_kf(x)\ \bigl(x\in\mathbb{R}^m \bigr), $$
    

holds for all \(y\in\mathbb{R}^{m}\) and k=1,…,m

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 \(n\in\mathbb{N}\).

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 non-periodic 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

Let Open image in new window and \(x\in\mathbb{R}\). Then the following holds for every A>0

$$I_A(f,x)=\int_{-A}^A \widehat{f}(y) e^{2\pi ixy}\,dy= \int_{-\infty}^{\infty} f(x-t)\frac{\sin2\pi At}{\pi t}\,dt. $$

Proof

Theorem

If functions Open image in new window and Open image in new window coincide in a neighborhood of a point x, then the convergence of the Fourier integral of f at x is equivalent to the convergence of the Fourier series of f ₀ at x, and, in the case of convergence, the following holds:

$$\int_{-\infty}^{\infty}\widehat{f}(y) e^{2\pi ixy}\,dy= \sum_{n=-\infty}^{\infty} \widehat{f}_0(n) e^{inx}. $$

Proof

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 one-dimensional case is a local property of a summable function, whereas the summability of the Fourier transform is a global property.

In contrast to the one-dimensional 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 multi-dimensional case (see Exercise 6).

The transformation that assigns the function Open image in new window defined by the formula

Open image in new window

to Open image in new window is called the inverse transform. Obviously, Open image in new window , and so the properties of the Fourier transform can easily be carried over to the inverse transform. Using the inverse transform, we can represent the inversion formula proved in the one-dimensional case in the following form:

Open image in new window

(5)

This justifies the choice of the term “inverse transform”.

Theorem

Let Open image in new window . If Open image in new window , then inversion formula (5) is valid for almost all x in \(\mathbb{R}^{m}\).

Proof

Using Eq. (6) and changing the order of integration, we obtain, for every t>0 that

$$\begin{aligned} (f*W_t) (x)=&\int_{\mathbb{R}^m}f(y)W_t(x-y) \,dy\\= &\int_{\mathbb{R}^m}f(y) \biggl(\int_{\mathbb{R}^m}e^{-\pi t^2\|u\|^2} e^{2\pi i \langle x-y,u\rangle}\,du \biggr)\,dy \\=&\int_{\mathbb{R}^m}e^{-\pi t^2\|u\|^2} e^{2\pi i\langle x,u\rangle} \biggl(\int _{\mathbb{R}^m}f(y) e^{-2\pi i\langle y,u\rangle}\,dy \biggr)\,du. \end{aligned}$$

Thus, we have established the required relation,

$$ (f*W_t) (x)=\int_{\mathbb{R}^m}e^{-\pi t^2\|u\|^2} e^{2\pi i\langle x,u\rangle} \widehat{f}(u)\,du. $$

(7)

Since the absolute value of the integrand in the last integral does not exceed \(|\widehat{f}|\), we obtain, by Lebesgue’s theorem, that, for every x, this integrals tends to Open image in new window as t→+0.

Now, we can finish the proof, referring to Theorem 10.3.4, from which it follows that the limit on the left-hand 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 left-hand 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

Corollary

If an even continuous function f is summable on the real line and is convex on the positive semi-axis, then f is the Fourier transform of a non-negative summable function.

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.

Theorem

If two finite Borel measures μ and ν have the same Fourier transform, then the measures coincide.

Proof

Let H _j (t)={(x ₁,…,x _m ) | x _j =t} be a plane perpendicular to the jth coordinate axis, and let

$$E=\bigl\{t\in\mathbb{R}\,|\,\mu\bigl(H_j(t)\bigr)=\nu \bigl(H_j(t)\bigr)=0\ \text{ for every }\ j=1,\ldots,m\bigr\}. $$

The set E is everywhere dense because the set \(\{t\in\mathbb{R}\,|\, \mu(H_{j}(t))>0\}\) is at most countable for each j (see Sect. 1.2.2). Therefore, the Borel hull of the semiring Open image in new window consisting of the cells whose vertices have coordinates belonging to E coincides with the σ-algebra of Borel subsets of the space \(\mathbb{R}^{m}\) (see the remark after Theorem 1.1.6). We express the measure of the cell \(P=\prod_{j=1}^{m}\Delta_{j}\) in terms of \(\widehat{\mu}\), assuming that Open image in new window .

Obviously, \(\chi_{P}(x)=\prod_{j=1}^{m}\chi_{\Delta_{j}}(x_{j})\), where x ₁,…,x _m are the coordinates of a vector x. By Fubini’s theorem, \(\widehat{\chi}_{P}(y)=\prod_{j=1}^{m}\widehat{\chi}_{\Delta_{j}}(y_{j})\) for y=(y ₁,…,y _m ), and, therefore,

$$I_A(\chi_{\Delta},x)=\int_{(-A,A)^m}\widehat{\chi}_P(y) e^{2\pi i\langle x,y \rangle}\,dy =\prod _{j=1}^m\int_{-A}^A \widehat{\chi}_{\Delta_j}(y_j) e^{2\pi ix_jy_j} \,dy_j. $$

The characteristic function of an interval satisfies Dini’s condition everywhere except the endpoints of the interval. Therefore, we have

$$\int_{-A}^A\widehat{\chi}_{\Delta_j}(y_j) e^{2\pi ix_jy_j}\,dy_j \underset{A\to+\infty}{\longrightarrow} \chi_{\Delta_j}(x_j) $$

for all j=1,…,m, provided that x _j is distinct from the endpoints of the interval Δ _j . Since Open image in new window , we see that \(I_{A}(\chi_{\Delta},x) \underset{A\to+\infty}{\longrightarrow}\chi_{P}(x)\) μ-almost everywhere. Moreover, putting Δ _j =[a _j ,b _j ), we obtain (see Lemma 10.5.3) that

$$\int_{-A}^A\widehat{\chi}_{\Delta_j}(y_j) e^{2\pi ix_jy_j}\,dy_j= \int_{-\infty}^{\infty} \chi_{\Delta_j}(x_j-t)\frac{\sin2\pi At}{t}\,dt= \int _{A(x_j-b_j)}^{A(x_j-a_j)}\frac{\sin2\pi u}{u}\,du. $$

All these integrals are bounded (since the integral \(\int_{0}^{\infty}\frac{\sin2\pi u}{u}\,du\) converges), and so, the integral I _A (χ _Δ,x) is also bounded (uniformly with respect to x and A). Therefore, we can use Lebesgue’s theorem on passing to the limit under the integral sign,

$$\begin{aligned} \mu(P)=&\int_{\mathbb{R}^m}\chi_P(x)\,d\mu(x)= \lim _{A\to+\infty}\int_{\mathbb{R}^m}I_A( \chi_{\Delta},x)\,d\mu(x) \\=&\lim_{A\to+\infty}\int_{\mathbb{R}^m} \biggl(\int _{(-A,A)^m}\widehat{\chi}_P(y) e^{2\pi i\langle x,y\rangle} \,dy \biggr)\,d\mu(x). \end{aligned}$$

Changing the order of integration, we obtain

$$\begin{aligned} \mu(P)=&\lim_{A\to+\infty}\int_{(-A,A)^m}\widehat{ \chi}_P(y) \biggl(\int_{\mathbb{R}^m}e^{2\pi i\langle x,y\rangle}\,d \mu(x) \biggr)\,dy\\=& \lim_{A\to+\infty}\int_{(-A,A)^m} \widehat{\chi}_P(y)\widehat{\mu}(-y)\,dy. \end{aligned}$$

This relation shows that the values of the measure on the cells belonging to Open image in new window can be expressed in terms of its Fourier transform. Since the measures μ and ν have the same Fourier transform, they coincide on the semiring Open image in new window , and, consequently, (by the uniqueness theorem) on all Borel sets. □

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 non-negative, 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 half-spaces (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

Let a function f in Open image in new window be orthogonal to all monomials. Obviously, f⊥P for every polynomial P in m variables. We put

$$F(y)=\int_{\mathbb{R}^m}f(x) e^{i\langle y,x\rangle}\,d\mu(x). $$

Since |e ^i〈y,x〉|≡1 and all polynomials are summable with respect to μ, the function F is infinitely differentiable and, for each y, the derivatives of F can be found by the Leibnitz rule.

We prove that F≡0. If ∥y∥<a/2, then expanding the exponential factor in a Taylor series and integrating termwise, we obtain that F(y)=0. The legitimacy of termwise integration follows from the fact that the partial sums of the series have a summable majorant, namely, |f(x)|e ^{∥x∥ ∥y∥} (this function is summable because the functions |f| and e ^{∥x∥ ∥y∥} belong to Open image in new window ). To prove that F≡0, we show that the interior G of the set where F(y)=0 coincides with \(\mathbb{R}^{m}\). Since G≠∅ (because it contains a neighborhood of zero), it is sufficient to verify that the set G is closed, in which case the equality \(G=\mathbb{R}^{m}\) will follow from the fact that the space \(\mathbb{R}^{m}\) is connected. Let \(y\in \overline{G}\). The function F and all its derivatives vanish at y by continuity. Calculating the derivatives by Leibnitz’s rule, we see that

$$0=F^{(n)}(y)=\int_{\mathbb{R}^m}f(x) (ix)^ne^{i\langle x,y\rangle} \,d\mu(x) \quad\bigl(n\in\mathbb{R}^m_+\bigr). $$

Thus, the function f ₁(x)=f(x)e ^i〈x,y〉 is orthogonal to all monomials. Replacing f by f ₁, we may assert by what has just been proved that the function \(F_{1}(\eta)=\int_{\mathbb{R}^{m}}f_{1}(x) e^{i\langle x,\eta\rangle}\,d\mu(x)\) assumes only zero values in a neighborhood of zero. However, F ₁(η) is nothing but F(y+η). Therefore, F≡0 in a neighborhood of y, i.e., y∈G. Thus, \(G=\overline{G}=\mathbb{R}^{m}\) and, consequently, F≡0. Now, we can easily complete the proof. Indeed, without loss of generality, we may assume that the function f is real. The identity F≡0 means that the measures f ₊ dμ and f ₋ dμ have the same Fourier transform. Consequently, these measures coincide by Theorem 10.5.5, which implies (by Theorem 4.5.4) that the functions f ₊ and f ₋ coincide almost everywhere with respect to μ. □

Corollary

The Hermite polynomials are complete in Open image in new window with \(d\mu(x)=e^{-x^{2}}\,dx\).

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 one-dimensional Lebesgue measure (for p⩾1 this effect is ruled out by the theorem just proved).

We will need the following formula from Example 1 of Sect. 7.1.7: if a>0 and z=e ^iθ , where \(\theta\in(0,\frac{\pi}{2})\), then \(z^{-a}\Gamma(a)=\int_{0}^{\infty} t^{a-1}e^{-z t}\,dt\). Comparing the imaginary parts and using the substitution t=x ^p /cosθ, we obtain

$$\begin{aligned} \Gamma(a)\sin a\theta=&\int_0^{\infty} t^{a-1}e^{-t\cos\theta}\sin(t\sin \theta)\,dt\\=& \frac{p}{ \cos^a \theta}\int_0^{\infty} x^{ap-1}e^{-x^p}\sin\bigl(x^p\text{tan}\, \theta \bigr)\,dx. \end{aligned}$$

Now, we use the freedom in the choice of the parameters a and θ. Putting \(\theta=\frac{\pi}{2}p\) and \(a=\frac{2}{p}(n+1)\), we obtain

$$\int_0^{\infty} x^{2n+1}e^{-x^p} \sin \biggl(x^p\text{tan}\,\frac{\pi }{2}p \biggr)\,dx=0 \quad \text{for}\ n=0,1,2\ldots. $$

This means that the odd function equal to \(\sin (x^{p}\text{tan}\,\frac{\pi}{2}p )\) for x⩾0 is orthogonal to all polynomials in the space Open image in new window with measure \(d\mu(x)=e^{- |x|^{p}}\,dx\).

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 .

First, we prove the assertion of the theorem for the smoothened function f _t . By properties of convolution, we have Open image in new window . By Theorem 10.5.1, we obtain \(\widehat{f}_{t}=\widehat{f}\widehat{\omega}_{t}\). This product is summable since the function \(\widehat{f}\) is bounded and Open image in new window by Corollary 10.5.2. Using Fubini’s theorem and inversion formula (5), we obtain

$$\begin{aligned} \int_{\mathbb{R}^m}\widehat{f}_t(y)\overline{ \widehat{f}_t(y)}\,dy=& \int_{\mathbb{R}^m} \widehat{f}_t(y)\overline{\biggl(\int_{\mathbb{R}^m}f_t(x) e^{-2\pi i\langle y,x\rangle}\,dx \biggr)}\,dy \\ =&\int_{\mathbb{R}^m}\widehat{f}_t(y) \biggl(\int _{\mathbb{R}^m}\overline{f_t(x)} e^{2\pi i\langle y,x\rangle}\,dx \biggr)\,dy \\ =&\int_{\mathbb{R}^m}\overline{f_t(x)} \biggl(\int _{\mathbb{R}^m}\widehat{f}_t(y) e^{2\pi i\langle y,x\rangle}\,dy \biggr)\,dx =\int_{\mathbb{R}^m}\overline{f_t(x)} \,f_t(x)\,dx. \end{aligned}$$

Thus,

$$ \|\widehat{f}_t\|^2_2=\|f_t\|^2_2. $$

(8)

It remains to verify that we can pass to the limit in this equation as t→0.

Since \(f_{t}\underset{t\to0}{\longrightarrow} f\) in the Open image in new window -norm, the continuity of the norm implies \(\|f_{t}\|_{2}\underset{t\to0}{\longrightarrow}\|f\|_{2}\). We verify that Open image in new window and \(\|\widehat{f}_{t}\|_{2}\underset{t\to0}{\longrightarrow}\|\widehat{f}\| _{2}\). To this end, we write the left-hand side of Eq. (8) in more detail,

$$ \|\widehat{f}_t\|_2^2=\int_{\mathbb{R}^m}\big|\widehat{f}_t(y)\big|^2\,dy= \int _{\mathbb{R}^m}\big|\widehat{f}(y)\big|^2\big|\widehat{\omega}_t(y)\big|^2\,dy. $$

(9)

Since \(\widehat{\omega}_{t}(y)\underset{t\to0}{\longrightarrow}1\) (see Corollary 7.6.3 with t ₀=0 and g(x)=e ^{−2πi〈y,x〉}), Fatou’s theorem and Eq. (8) imply

$$\begin{aligned} \int_{\mathbb{R}^m}\big|\widehat{f}(y)\big|^2\,dy\leqslant&\lim _{t\to0}\int_{\mathbb{R}^m}\big| \widehat{f}(y)\big|^2\big| \widehat{\omega}_t(y)\big|^2\,dy=\lim_{t\to0} \|\widehat{f}_t\|_2^2= \lim _{t\to0}\|f_t\|_2^2=\|f \|_2^2\\ <&+\infty. \end{aligned}$$

Thus, Open image in new window . Returning to Eq. (9), we see that the integrand in the integral on the right has a summable majorant, namely, \(|\widehat{f}|^{2}\). Therefore, we can pass to the limit in this integral by Lebesgue’s theorem,

$$\int_{\mathbb{R}^m}\big|\widehat{f}(y)\big|^2 \big|\widehat{\omega}_t(y)\big|^2\,dy \underset{t\to0}{\longrightarrow} \int_{\mathbb{R}^m}\big| \widehat{f}(y)\big|^2\,dy. $$

Now, the passage to the limit in Eq. (8) leads to the required result. □

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

From Plancherel’s theorem, it follows that the sequence \(\{ \widehat{f}_{n}\}_{n\geqslant1}\) is fundamental,

$$\|\widehat{f}_n-\widehat{f}_k\|_2=\| \widehat{f_n-f_k}\|_2=\|f_n-f_k \| _2\underset{n,k\to\infty}{\longrightarrow}0. $$

The limit exists because the space Open image in new window is complete (see Theorem 9.1.3). If {g _n } _n⩾1 is another sequence of functions in Open image in new window convergent to f in the Open image in new window -norm, then the sequence f ₁,g ₁,f ₂,g ₂,… obtained by “shuffling” the sequences {f _n } _n⩾1 and {g _n } _n⩾1 converges to f. By what has just been proved, the sequence \(\widehat{f}_{1},\widehat{g}_{1},\widehat{f}_{2}, \widehat{g}_{2},\ldots\) has a limit, which is unique up to equivalence and coincides with the limits of its subsequences. □

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 \(\|f_{n}-f\|_{2}\underset{n\to\infty}{\longrightarrow}0\).

Theorem

Let Open image in new window . Then:

1. (1)

   \(\|\widehat{f}\|_{2}=\|f\|_{2}\);

    
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. (3)

   we have Open image in new window ;

    
4. (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, \(\|\widehat{f}_{n}-\widehat{f}\|_{2}=\|\widehat{f_{n}-f}\|_{2}=\| f_{n}-f\|_{2} \underset{n\to\infty}{\longrightarrow}0\).