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Fourier Transforms (II)

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Fourier Analysis of Economic Phenomena

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

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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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Notes

  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

    Takagi [13] p. 166.

  5. 5.

    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.

  6. 6.

    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.

  7. 7.

    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.

  8. 8.

    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.

  9. 9.

    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 (†).

  10. 10.

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

  11. 11.

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

  12. 12.

    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.

  13. 13.

    cf. Yosida and Kato [15] pp. 98–99.

  14. 14.

    cf. Yosida [16] pp. 151–155.

  15. 15.

    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 \(\mathfrak {L}^2(\mathbb {R},\mathbb {C})\).

  16. 16.

    This section is based upon Maruyama [10].

  17. 17.

    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.

  18. 18.

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

  19. 19.

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

  20. 20.

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

  21. 21.

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

  22. 22.

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

  23. 23.

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

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  1. Professor Emeritus, Keio University, Tokyo, Japan

    Toru Maruyama

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Maruyama, T. (2018). Fourier Transforms (II). In: Fourier Analysis of Economic Phenomena. Monographs in Mathematical Economics, vol 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-2730-8_4

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