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2018 | OriginalPaper | Buchkapitel

4. Fourier Transforms (II)

verfasst von : Toru Maruyama

Erschienen in: Fourier Analysis of Economic Phenomena

Verlag: Springer Singapore

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Abstract

In the previous chapter, we observed a peculiar relation between the smoothness and the rapidity of vanishing at infinity of a function f, as well as its Fourier transform $\hat {f}$. Based upon this observation, we introduce an important function space $\mathfrak {S}$, which is invariant under the Fourier transforms. We then proceed to $\mathfrak {L}^2$-theory of Fourier transforms due to M. Plancherel. As a simple application of Plancherel’s theory, we discuss how to solve integral equations of convolution type. Finally, a tempered distribution is defined as an element of $\mathfrak {S}'$, and its Fourier transform is examined in detail.

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Vorheriges Kapitel Fourier Transforms (I)

Nächstes Kapitel Summability Kernels and Spectral Synthesis

Where the domain and the range are clear enough, we write just $\mathfrak {S, D}$ and so on, for brevity.

SeeSchwartz [11],Bourbaki [1] andGrothendieck [4] for the theory of locally convex spaces. See also Appendix B in this book.

A more general theorem is explained in Maruyama [9] pp. 232–233.

Takagi [13] p. 166.

The proof of (4.7) here is due to Yosida [16], p. 147. Although this approach is a little bit technical, I adopt it because of its simplicity. There are various other approaches. For instance, see Kawata [6] Chap. 11, Treves [14] Theorem 25.1 or Maruyama [8] Theorem 4.13.

A direct proof is also possible. Since $f \in \mathfrak {S}$ is bounded and integrable, f ⋅ f is integrable (i.e. $f \in \mathfrak {L}^2$) by Hölder’s inequality. We have only to approximate $f \in \mathfrak {L}^2$ by a simple function φ with compact support and to approximate φ by a smooth function with compact support.

The proof here is due to Dunford–Schwartz [2] III, pp. 1988–1989. See also Sect. 4.4 in this chapter and Schwartz [12] Chap. VII for the connection with the theory of distributions.

In general, it holds good that $u\ast v\in \mathfrak {L}^p(\mathbb {R}, \mathbb {C})$ and $\|u\ast v\|{ }_p\leqq \|u\|{ }_p \cdot \|v\|{ }_1$ for any $u\in \mathfrak {L}^p(\mathbb {R}, \mathbb {C})\, (1\leqq p\leqq \infty )$ and $v\in \mathfrak {L}^1(\mathbb {R}, \mathbb {C})$. See Maruyama [9] pp. 235–236.

The relation (4.19) can be verified as follows. In general, if $u_r\in \mathfrak {L}^2(\mathbb {R},\mathbb {C})$, and $v \in \mathfrak {L}^1 (\mathbb {R},\mathbb {C})$, and $\underset {r\to \infty }{\mathrm {l.i.m.}}u_r=u$, then

$$\displaystyle \begin{aligned} \underset{r \to \infty}{\mathrm{l.i.m.}}\int v(x-z)u_r(z)dz=\int v(x-z)u(z)dz. \end{aligned} $$

(†)

In fact, by the footnote just above,

$$\displaystyle \begin{aligned} \bigg\|\int v(x-z)u_r(z)dz-\int v(x-z)u(z)dz\bigg\|{}_2 &=\bigg\| \int [u_r(z)-u(z)]v(x-z)dz\bigg\|{}_2 \\ &\leqq \| u_r-u\|{}_2\cdot \|v\|{}_1 \to 0 \quad \text{as}\quad r \to \infty, \end{aligned} $$

which establishes (†). Now (4.19) immediately follows from (†).

This section is basically due to Kawata [5]II, Chap. 17. However, the proof of (4.19) given there does not seem correct.

cf. Appendix B, Sect. B.2 (p. 366).

While the Fourier transform on $\mathfrak {S}(\mathbb {R})$ is denoted by $\mathcal {F}$, the Fourier transform on $\mathfrak {S}(\mathbb {R})'$ is denoted by the German Fraktur letter $\mathfrak {F}$. The different letters are used for the sake of clear distinction.

cf. Yosida and Kato [15] pp. 98–99.

cf. Yosida [16] pp. 151–155.

Since $f\in \mathfrak {L}^2(\mathbb {R},\mathbb {C})$ determines a tempered distribution T _f, it has its inverse $\widetilde {T_f}\in \mathfrak {S}(\mathbb {R})'$. $\widetilde {T_f}$ is a continuous linear functional on $\mathfrak {S}(\mathbb {R})$ (with respect to $\mathfrak {L}^2$-norm). This can be checked by a similar computation to that in (4.30). Hence $\widetilde {T_f}$ can be uniquely extended to a continuous linear functional on $\mathfrak {L}^2(\mathbb {R},\mathbb {C})$. By Riesz’s theorem, there exists some $\tilde {f}\in \mathfrak {L}^2(\mathbb {R},\mathbb {C})$ which represents $\widetilde {T_f}$. $\tilde {f}$ is given in a concrete form

$$\displaystyle \begin{aligned} \tilde{f}(x)=\underset{h\rightarrow \infty}{\mathrm{l.i.m.}}\frac{1}{\sqrt{2\pi}} \int_{|y|\leqq h} e^{ixy}f(y)dy \end{aligned} $$

(cf. (4.36) and (4.37)). $\widetilde {T_f}=T_{\tilde {f}}$ is the inverse operator of $\widehat {T_f}=T_{\hat {f}}$ on $\mathfrak {S}(\mathbb {R})$. So it is clear that these are mutually inverse as operators extended to $\mathfrak {L}^2(\mathbb {R},\mathbb {C})$.

This section is based upon Maruyama [10].

We denote by $\mathfrak {L}_{loc.}^1(\mathbb {R},\mathbb {C})$ the space of locally integrable complex-valued functions defined on $\mathbb {R}$. $\mathfrak {D}(\mathbb {R})$ is the space of test functions. See Sect. 4.4 and Appendix C.

$\mathfrak {D}(\mathbb {R})'$ denotes the dual space of $\mathfrak {D}(\mathbb {R})$. Each element of $\mathfrak {D}(\mathbb {R})'$ is called a distribution.

The notation supp θ means the support of the function θ.

$\displaystyle \sum _{k=-p}^p\theta (x+2k\pi ) \rightarrow \sum _{n=-\infty }^\infty \theta (x+2n\pi )$ (in $\mathfrak {C}^\infty $) on supp ψη.

We should note that supp ψ _α θ ⊂supp θ.

See Folland [3], pp. 320–322 and Lax [7], p. 570 for an outline of ideas.

n ≠ n _r ⇒ λ(n − n _r) = 0, n = n _r ⇒ λ(n _r − n _r) = 1.

Bourbaki, N.: Eléments de mathématique; Espaces vectoriels topologiques. Hermann, Paris, Chaps. 1–2 (seconde édition) (1965), Chaps. 3–5 (1964) (English edn.) Elements of Mathematics; Topological Vector Spaces. Springers, Berlin/Heidelberg/New York (1987)

Dunford, N., Schwartz, J.T.: Linear Operators, Part 1–3. Interscience, New York (1958–1971)MATH

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

Grothendieck, A.: Topological Vector Spaces. Gordon and Breach, New York (1973)MATH

Kawata, T.: Ohyo Sugaku Gairon (Elements of Applied Mathematics). I, II, Iwanami Shoten, Tokyo (1950, 1952) (Originally published in Japanese)

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

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

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

Maruyama, T.: Sekibun to Kansu-kaiseki (Integration and Functional Analysis). Springer, Tokyo (2006) (Originally published in Japanese)

10.

Maruyama, T.: Herglotz-Bochner representation theorem via theory of distributions. J. Oper. Res. Soc. Japan 60, 122–135 (2017)MathSciNetMATH

11.

Schwartz, L.: Functional Analysis. Courant Institute of Mathematical Sciences, New York University, New York (1964)

12.

Schwartz, L.: Théorìe des distributions. Nouvelle édition, entièrement corrigée, refondue et augmentée, Hermann, Paris (1973)

13.

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

14.

Treves, J.F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)MATH

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Yosida, K., Kato T.: Ohyo Sugaku (Applied Mathematics). I, Shokabou, Tokyo (1961) (Originally published in Japanese)

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

Titel: Fourier Transforms (II)
verfasst von: Toru Maruyama
Verlag: Springer Singapore
Buch: Fourier Analysis of Economic Phenomena
Print ISBN: 978-981-13-2729-2

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

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-981-13-2730-8_4

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