Classical Fourier Transforms pp 91-135 | Cite as

# Fourier transforms on L_{2}(-∞, ∞)

Chapter

## Abstract

The Banach space L, the integral existing because of Schwarz’s inequality.

_{2}(-∞,∞) is endowed with an*inner product*. For any two functions f,g belonging to L_{2}(-∞,∞), the inner product (f,g) is defined by$$(\textup{f, g})=\int_{-\infty} ^{\infty} \textup{f(x)}\bar{\textup{g}}\textup{(x)dx}$$

## Keywords

FOURIER Transform Fourier Series Entire Function Fourier Coefficient Dense Subset
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## Notes

- §2.§3._Different proofs of Plancherel’s [1] theorem have been given by Titchmarsh [2], Bochner [1], F. Riesz [1], Wiener [3], M.H. Stone [1], p.104, and Bochner and Chandrasekharan [1]. For the Remarks following (2.29) see Stein and Weiss [1], p.18. For (3.5) see Bochner and Chandrasekharan [1], p.99.Google Scholar
- §4.Theorem 6 is due to Wiener; for Theorem 7 see Bochner and Chandrasekharan [1], Ch.IV, §10.Google Scholar
- §5.Weyl’s proof of the inequality under somewhat stronger hypotheses is given in Appendix I to his book [2]. His proof in the second edition differs in detail from the one given in the first. The proof given here differs only in detail from the one given in their book by Dym and McKean [1].Google Scholar
- §6.The Phragmén-Lindelöf [1] principle takes many forms. See, for instance, Calderón-Zygmund [1], Littlewood [2], p.107, Titchmarsh [2], §5.71. For Hardy’s theorem, see Hardy [1], and Titchmarsh [3], p.174, where further references are given.Google Scholar
- §7.Paley and Wiener [1] were the first to make a systematic study of Fourier transforms in the complex domain (one variable). The proof given here differs only in detail from the one presented by Dym and McKean [1], Ch.3. For functions of exponential type see, for instance, Boas [1].Google Scholar
- §8.For generalities on Fourier orthogonal series see Kaczmarz and Steinhaus [1], Ch.II, where several examples of orthogonal systems are given, including Rademacher’s [1], and Walsh’s [1] which can be defined, after Paley [1], in terms of Rademacher’s functions. Let χ
_{o}(t) =1, and if N is a positive integer, expressed in the binary scale as, with, then, where the (ϕ_{n}) are Rademacher’s functions. The system (χ_{n}) is ortho-normal over (0,1), and complete, and is known as Walsh’s. Bessel’s inequality (in several variables) has been shown by Siegel [1] to yield Minkowski’s first theorem on lattice points in convex sets [cf. the author’s book [2], p.99]. Atle Selberg [1] has shown that Bombieri’s large sieve inequality can be viewed as a form of Bessel’s inequality in a Hilbert space, cf. H.L. Montgomery [1]. For a concise introduction to Fourier trigonometric series in L_{2}(0,2π) see Hardy and Rogosinski [1]. Series and integrals can be treated together on a group space. See Butzer and Nessel [1]. For Lebesgue’s proof of the completeness of the trigonometric system, see Hardy and Rogosinski [1], or Zygmund [1], Vol.1, Ch.I, §6.Google Scholar - §9.Hardy’s [2] interpolation formula is also treated in Zygmund [1], Vol.11, p.276. As Dr. Albert Stadler has remarked, the condition of boundedness in Theorem 12 can be replaced by one of polynomial growth, in which case formula (9.7) will assume a more general form.Google Scholar
- §10.S. Bernstein’s work [1] is also presented in Zygmund [1], Vol. II, p.11, Ch.X, and p.276, Ch.XVI. Zygmund’s inequality for the integrated derivative of a trigonometric polynomial, as a generalization of Theorem 14, is given by him immediately after Bernstein’s result. N.G. de Bruijn [1] has given generalizations of Bernstein’s theorem for polynomials in the complex domain. For the Remark preceding Theorem 14, see Siegel [1]. See also Stein [2].Google Scholar
- §11.For the extension of the Paley-Wiener theorem to k dimensions, k> 1, see Plancherel and Pólya [1], Stein [2], Stein and Weiss [1], Ch.III, Th.4.9. The last-mentioned reference connects the theorem with the analysis of H
^{p}-spaces. See Narasimhan [1], Ch.3, for the rôle of the Fourier transform in analytical problems on manifolds; also Ehrenpreis [1] in connexion with several complex variables.Google Scholar

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