Convergence of Classical Fourier Series

Maruyama, Toru

doi:10.1007/978-981-13-2730-8_2

Convergence of Classical Fourier Series

Toru Maruyama⁶

Chapter
First Online: 03 July 2019

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Part of the book series: Monographs in Mathematical Economics ((MOME,volume 2))

Abstract

We have discussed basic contents of the theory of Fourier series on a general Hilbert space. We now proceed to the classical problem concerning the Fourier series expansion of an integrable function with respect to the trigonometric functions. If we choose $\mathfrak {L}^2([-\pi , \pi ], \mathbb {C})$ as a Hilbert space and

$$\displaystyle \frac {1}{\sqrt {2\pi } }, \frac {1}{\sqrt {\pi }} \cos x , \; \frac {1}{\sqrt {\pi } } \sin x , \; \cdots ,\; \frac {1}{\sqrt {\pi } } \cos nx , \; \frac {1}{\sqrt {\pi } } \sin nx , \; \cdots ; \; n=1,2, \cdots $$

as a complete orthonormal system, the Fourier series of $f\in \mathfrak {L}^2([-\pi , \pi ],\mathbb {C})$ is given in the form

$$\displaystyle \frac {a_0}{2} + \sum _{n=1}^\infty (a_n \cos nx + b_n \sin nx) , $$

where

$$\displaystyle a_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \cos nx \; dx ,\quad b_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \sin nx \; dx . $$

This Fourier series converges to f in $\mathfrak {L}^2$-norm.

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Notes

1.
We owe this result to Kolmogorov– Fomin [14] pp. 387–388. For more general results, see Kawata [11] pp. 63–64.
2.
The requirement (2.14) is called the Dini condition in Kolmogorov–Fomin [14], Vol.II, p. 388.
3.
Weierstrass-type. Let $g[a,b]\rightarrow \mathbb {R}$ be integrable and $h: [a,b]\rightarrow \mathbb {R}$ be bounded and monotone. Then there exists some η ∈ [a, b] such that
$$\displaystyle \begin{aligned} \int_{a}^{b} g(x)h(x)dx = h(a) \int_{a}^{\eta } g(x) dx + h(b) \int_{\eta }^{b} g(x) dx \end{aligned}$$

(cf. Stromberg [16] pp. 328–329, Takagi [17] pp. 287–288).
Bonnet-type. Let $g: [a,b]\rightarrow \mathbb {R}$ be integrable and $h: [a,b] \rightarrow \mathbb {R}$ be nonnegative and nonincreasing. Then there exists some η ∈ [a, b] such that
$$\displaystyle \begin{aligned} \int_{a}^{b} g(x)h(x)dx = h(a) \int_{a}^{\eta } g(x) dx. \end{aligned}$$

If h is nonnegative and nondecreasing, then there exists some η ∈ [a, b] such that
$$\displaystyle \begin{aligned} \int_{a}^{b} g(x) h(x) dx = h(b) \int_{\eta }^{b} g(x) dx \end{aligned}$$

(cf. Kawata [10] I, p. 18, Stromberg [16] p. 334).
4.
We acknowledge Kawata [11] pp. 83–84 for the proof. It can be proved that satisfies (2.15). See fine calculations by Kawata [10] I, pp. 67–69.
5.
The proof here is due to Kawata [11] p. 85.
6.
Kolmogorov– Fomin [14] II, pp. 390–391.
7.
Banach–Steinhaus resonance theorem: Let $\mathfrak {X}$ be a Banach space, $\mathfrak {Y}$ a normed space, and {T _α|α ∈ A} a family of bounded linear operators of $\mathfrak {X}$ into $\mathfrak {Y}$. If {T _α x|α ∈ A} is bounded in $\mathfrak {Y}$ for each $x\in \mathfrak {X}$, then {T _α|α ∈ A} is uniformly bounded, i.e. sup_{α ∈ A}∥T _α∥ < ∞. (The converse is obvious.) cf. Dunford– Schwartz [2] pp. 52–53, Maruyama [15] pp. 344–345, Yosida [18] p. 69.
8.
The classical works cited here are Hardy [3], Kolmogorov [12, 13], Katznelson [9] and Kahane–Katznelson [7]. See also Hardy–Rogozinski [4]. Zygmund [19] is the best treatise in the discipline.
9.
The classical works are Carleson [1] and Hunt [5]. Jørsboe and Melbro [6] devoted their entire volume to a clear-cut proof of Carleson’s theorem.
10.
Let u be absolutely continuous and v be integrable on [a, b]. Then u, v is also integrable. Denoting by V the indefinite integral of v, we obtain the formula:
$$\displaystyle \begin{aligned} \int_a^b u(t)v(t)dt=u(b)V(b)-u(a)V(a)-\int_a^b u'(t)V(t)dt. \end{aligned}$$

cf. Kato [8] p. 107, Stromberg [16] p. 323.
11.
cf. Stromberg [16] pp. 141–142, Takagi [17] p. 155, Theorem 39.
12.
The proof given here is due to Kolmogorov– Fomin [14], pp. 391–392.
13.
Takagi [17] p. 9, p. 275. Related results are neatly discussed in Stromberg [16] pp. 473–484.

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Professor Emeritus, Keio University, Tokyo, Japan
Toru Maruyama

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Maruyama, T. (2018). Convergence of Classical Fourier Series. In: Fourier Analysis of Economic Phenomena. Monographs in Mathematical Economics, vol 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-2730-8_2

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DOI: https://doi.org/10.1007/978-981-13-2730-8_2
Published: 03 July 2019
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2729-2
Online ISBN: 978-981-13-2730-8
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