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2018 | OriginalPaper | Chapter

1. Fourier Series on Hilbert Spaces

Author : Toru Maruyama

Published in: Fourier Analysis of Economic Phenomena

Publisher: Springer Singapore

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Abstract

Let e ₁, e ₂, …, e _l be the standard basis of an l-dimensional Euclidean space consisting of l unit vectors. Then any vector x can be expressed as

$$\displaystyle x=\sum _{i=1}^l c_ie_i $$

and such an expression is determined uniquely. The coefficients c ₁, c ₂, ⋯, c _l are computed as c _i = 〈x, e _i〉 (inner product).

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next chapter Convergence of Classical Fourier Series

A systematic theory of functional analysis in the framework of Hilbert spaces is discussed in many textbooks in this discipline. For instance, Halmos [4] and Schwartz [8] are very well-written classics. See also Lax [6] Chap. 6 and Maruyama [7] Chap. 3.

In the case of a real vector space $\mathfrak {H}$, an inner product is a real-valued function $\langle \cdot, \cdot \rangle : \mathfrak {H}\times \mathfrak {H}\rightarrow \mathbb {R}$ such that (i)–(iv) are satisfied. Of course, (ii) should be rewritten as 〈x, y〉 = 〈y, x〉.

See Folland [3] Chap. 6, Terasawa [10] pp. 145–149, pp. 414–416, Yosida [11] Chap. 1, §3 and Chap. 2, §2.

The term of the highest degree = (1∕2ⁿ n!){(2n)(2n − 1)⋯(n + 1)}x ⁿ.

The first process of integration by parts is as follows:

$$\displaystyle \begin{aligned}f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n \Big|{}_{-1}^1-\int_{-1}^1 f'(x)\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n dx \end{aligned}$$

$$\displaystyle \begin{aligned}=-\int_{-1}^1f'(x)\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^{n}dx.\end{aligned}$$

We denote by $\mathbb {C}_+$ the complex half-plane $\mathbb {R}ez>0$. The function $(\varGamma : \mathbb {C}_+\rightarrow \mathbb {C})$ defined by

$$\displaystyle \begin{aligned} \varGamma(s)=\int_0^\infty x^{s-1}e^{-x}dx \end{aligned} $$

(t)

is called the gamma function. Γ is analytic on $\mathbb {C}_+$. Sometimes the domain of Γ is taken to be (0, ∞) rather than $\mathbb {C}_+$. In the text above, we also follow this policy. The function $B: \mathbb {C}_+\times \mathbb {C}_+\rightarrow \mathbb {C}$ defined by

$$\displaystyle \begin{aligned} B(p,q)=\int_0^1 x^{p-1}(1-x)^{q-1}dx \end{aligned}$$

is called the beta function. (Note that the integral is convergent.) The following formulas hold good (cf. Cartan [1] Chap. V, §3 and Takagi [9] Chap. 5, §68):

1∘

Γ(s + 1) = sΓ(s).

We also obtain Γ(n + 1) = n! for all $n\in \mathbb {N}$ by induction.

2∘

B(p, q) = Γ(p)Γ(q)∕Γ(p + q).

3∘

$\varGamma (s)\varGamma (1-s)=\pi /\sin \pi s.$

If s = 1∕2, in particular, we have $\varGamma (1/2)=\sqrt {\pi }$.

4∘

$\varGamma \big (n+\frac {1}{\, 2\,}\big )=\frac {1}{\,2\,}\big (\frac {3}{\, 2\, }\big )\cdots \big (n-\frac {1}{\, 2\,}\big )\sqrt {\pi } \quad (n=0,1,2,\cdots ).$

More rigorously, the Fourier coefficient corresponding to $(1/\sqrt {2\pi })e^{inx}$ is

$$\displaystyle \begin{aligned}\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi f(x)e^{-inx}dx,\end{aligned}$$

and the Fourier series is

$$\displaystyle \begin{aligned}\sum_{n=-\infty}^\infty\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi f(x)e^{-inx}dx\cdot\frac{1}{\sqrt{2\pi}} e^{inx}. \end{aligned}$$

See Folland [3] Chap. 6 and Kawata [5] pp. 33–36 for other orthonormal systems.

We acknowledge Yosida [12] p. 88 for the proof here.

The injectivity of T is also verified by Theorem 1.6 (ii).

Cartan, H.: Théorie élémentaires des fonctions analytiques d’une ou plusieurs variables complexes. Hermann, Paris (1961) (English edn.) Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison Wesley, Reading (1963)

Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brooks, Pacific Grove (1988)

Folland, G.B.: Fourier Analysis and its Applications. American Mathematical Society, Providence (1992)

Halmos, P.R.: Introduction to a Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York (1951)

Kawata, T.: Fourier Kaiseki (Fourier Analysis). Sangyo Tosho, Tokyo (1975) (Originally published in Japanese)

Lax, P.D.: Functional Analysis. Wiley, New York (2002)

Maruyama, T.: Kansu Kaisekigaku (Functional Analysis). Keio Tsushin, Tokyo (1980) (Originally published in Japanese)

Schwartz, L.: Analyse hilbertienne. Hermann, Paris (1979)

Takagi, T.: Kaiseki Gairon (Treatise on Analysis), 3rd edn. Iwanami Shoten, Tokyo (1961) (Originally published in Japanese)

10.

Terasawa, K.: Shizen Kagakusha no tame no Sugaku Gairon (Treatise on Mathematics for Natural Scientists). Iwanami Shoten, Tokyo (1954) (Originally published in Japanese)

11.

Yosida, K.: Lectures on Differential and Integral Equations. Interscience, New York (1960)

12.

Yosida, K.: Functional Analysis, 3rd edn. Springer, New York (1971)CrossRef

Title: Fourier Series on Hilbert Spaces
Author: Toru Maruyama
Publisher: Springer Singapore
Book: Fourier Analysis of Economic Phenomena
Print ISBN: 978-981-13-2729-2

Electronic ISBN: 978-981-13-2730-8

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-981-13-2730-8_1

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