Fourier transforms on L1(-∞, ∞)
Part of the Universitext book series (UTX)
We assume as known Lebesgue’s theory of integration.
KeywordsFourier Transform Banach Algebra Inversion Formula Composition Rule Property Versus
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- §1.Theorem 1 is a generalization of the basic result on Fourier series, which states that the Fourier coefficients of an integrable function tend to zero, which was proved by Riemann for Riemann-integrable functions, and extended by Lebesgue to Lebesgue-integrable functions. The idea of the second proof sketched in the Remarks following Theorem 1 is due to Lebesgue, Bull. Societe Math, de France, 38(1910), 184–210. A peculiar generalization of Theorem 1 has been given by Bochner and Chandrasekharan in , Th.46, Ch.III. See the later definition of pseudo-characters by Bochner  Ch.3, p.53. A characteristic function in the sense used here is not the same as in Chapter III, hence the alternative term “indicator function”. For the rôle of Hermite functions in Fourier analysis, see, for instance, N. Wiener . The standard work on Bessel functions is Watson’s . A short introduction is given in Ch.XVII of Whittaker and Watson . Example 10 is due to S. Ramanujan, J. Indian Math. Soc. 11 (1919), 81–87; Coll. Papers, No. 23 . The integral in Step (i) is evaluated by Cauchy’s theorem in Lindelöf’s book ; see p. 49 for the motivation of the proof.Google Scholar
- §2.For the notion of an algebra, and basic facts about algebras, see, for instance, G. Birkhoff and S. MacLane: “A survey of modern algebra”, p. 225. For an introduction to Fourier analysis on groups, see the classic by Weil ; also Loomis , Naimark , the Appendix in Goldberg , Rudin , and Reiter , where the contributions of A. Beurling, I.E. Segal, and others, are described.Google Scholar
- §3.For the theory of distributions, in general, with applications, see the classics by L. Schwartz , and I.M. Gelfand and G.E. Shilov . For distributions in connexion with Fourier transforms, in particular, see Ch.I of Hörmander’s book , also Yosida , Ch.VI, and Donoghue . The function ω in (3.12) was introduced by Wiener , p.562.Google Scholar
- §4.The localization theorem here is motivated by the one on Fourier series due to Riemann, see Hardy and Rogosinski , pp. 39–42, and Zygmund , Ch.II, §6, p. 53, §8. The examples of Mellin transforms given here are of frequent occurrence in analytic number theory, see, for instance, the author’s book . Several more are given by Titchmarsh .Google Scholar
- §5.For Poisson’s summation formula, see Bochner ; his proof is also given in the author’s book . For some special applications see, for instance, Zygmund , Vol.1, Ch.II, §13. For the theta-relation (5.8), in the general setting of theta-functions, see, for instance, the author’s book , where the connexion with elliptic functions, and the theory of numbers, is elucidated on an elementary level. There is also an L2-version in one variable, see Boas .Google Scholar
- §6.The proof of the uniqueness in Theorem 7 (without the use of summability and general inversion) can be effected by the use of a piece-wise linear (trapezoidal) function instead of the function ωC,ε; see Bochner and Chandrasekharan , Ch.I, §6, Th.5. For a sharper version of the uniqueness theorem due to A.C. Offord , see Th.10, Ch.III.Google Scholar
- §7.The motivation for the summability theorems here is again supplied by the theory of Fourier series, see, for instance, Hardy and Rogosinski , Ch.V, p. 70; Bochner and Chandrasekharan , §7. For properties (7.6) and (7.20) of integrable functions, and for the definition of the “Lebesgue set”, see, for instance, Titchmarsh , §11.6. Convolution integrals of the type (7.1) are of importance in the theory of approximation. See Butzer and Nessel , where generalized singular integrals of the type of Cauchy-Poisson, Gauss-Weierstrass, Fejér, and Bochner-Riesz, are dealt with in detail. For more general methods, see Stein , Stein and Weiss .Google Scholar
- §8.Example 3, following Theorem 11’, is used by C.L. Siegel in his proof of Hamburger’s theorem on the Riemann zeta-function. See, for instance, the author’s book , Ch.II, §5. Theorem 12 is due to Bochner and Chandrasekharan , Ch.I, Th.9, p.20; also p.211, where it is commented upon. A generalization was later given by Bochner in his book , p.25, Th.2.2.1. Theorems 13 and 14 make it possible to define the Fourier transform on L2 (-∞,∞) and prove Plancherel’s theorem [cf. Ch.II] by starting from the subspace L1·∩·L2.Google Scholar
- §9.The systematic use of summability in norm seems to have originated with Wiener. For Theorem 17 see Bochner and Chandrasekharan , Ch.I, §10, who also proved further results in that direction. On Weyl’s form of the Riesz-Fischer theorem, see Weyl , and Wiener’s  remarks; also Stone’s , p.26; and J. von Neumann’s , p. 109–111.Google Scholar
- §10.For the central limit theorem, see Cramér , Ch.17, §4, and Feller , Ch.VIII, §4. According to Cramér, the theorem was first stated by Laplace in 1812; a rigorous proof under “fairly general” conditions was given by Liapounoff; and the problem of finding the most general conditions of validity was solved by Feller, Khintchine, and Lévy. The proof given here differs only in detail from that given, for instance, by Dym and McKean , Ch.2, §7.Google Scholar
- §11.Theorem 21 is the analogue, for Fourier transforms, of a classical theorem on the absolute convergence of Fourier series due to N. Wiener  and P. Lévy . See Zygmund , Vol.1, Ch.VI, §5. The proof given here differs only in detail from that of R.R. Goldberg , Ch.2, §9, which is itself closely modelled on Bochner’s proof  of the Wiener-Lévy theorem.Google Scholar
- §12.Wiener was the first to study “closure” properties of functions in L1 (-∞,∞) and in L2(-∞,∞), and relate them to Fourier transform theory. See Wiener . The proof given here of Theorem 23 is the same as Bochner’s . An algebraic reformulation of the theorem would be that every proper closed ideal of L1(-∞,∞) is contained in a maximal ideal. The problem of characterizing the sub-class of functions f in L1(-∞,∞) which have the property that Sf is the intersection of the maximal ideals containing it has received attention. For the work of Beurling and others, see Pollard , and Reiter . For generalizations of Theorem 23, see Ch.4 of Goldberg , Reiter , where further references can be found, e.g. to Agmon and Mandelbrojt , Malliavin , and others.Google Scholar
- §13.Theorems 24 and 25 are due to Wiener . Theorem 26 is due to Littlewood , and forms the prototype for many of the tauberian theorems of the period before Wiener. Littlewood’s theorem can be proved directly, and simply, as Karamata  has shown, by the use of Weierstrass’s theorem on the approximation of continuous functions, instead of Wiener’s theorem on the L1-closure. See Wiener’s own remarks , and Wielandt’s  arrangement of Karamata’s proof. Wiener’s work on tauberian theorems has been carried forward notably by H.R. Pitt . Albert Stadler  has recently proved a tauberian theorem, with remainder, of the Wiener-Ikehara type (see, for instance, the author’s book ), which yields the more refined forms of the prime number theorem as corollaries. See Wiener’s remarks , p.93, on this possibility. Bochner and Chandrasekharan , Th.29, p.54, subsume Karamata’s theorem as part of another theorem which characterizes what they call the Karamata extension of the kernel e-α. The nature of this extension in the case of general kernels seems not to be known.Google Scholar
- §14.Theorems 27 and 28 are due to Bochner and Chandrasekharan , Ch.I, §15. See Butzer and Nessel  Ch.7, for later developments. Equations (14.1) and (14.2) arise in connexion with the problem of conduction of heat, see Carslaw , §§16, 45.Google Scholar
- §15.The proof of (15.24) given here is the same as the one given by Bochner in his book ; he comments that according to Burkhardt  pp. 1165–1173, the cases k = 2,3 are due to Poisson and Cauchy, and that the “theorem is also not new for k arbitrary”. A second proof is given by Bochner and Chandrasekharan , pp.71–74. The introduction of the spherical mean fx(t) is due to Bochner. He carried the idea further into the study of multiple Fourier series. See Bochner , followed by Chandrasekharan , Chandrasekharan and Minakshisundaram , and , Ch.IV, and H. Joris . Important work with quite different techniques has been accomplished on topics in multiple Fourier series by E.M. Stein, and others. See, for instance, Ch.VII of Stein and Weiss . The evaluation of Vk(s) by induction is done, for instance, by Walfisz , p.41.Google Scholar
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