Classical Fourier Transforms pp 137-159 | Cite as

# Fourier-Stieltjes transforms (one variable)

Chapter

## Abstract

We assume as known the fundamentals of the theory of Riemann-Stieltjes integrals.

## Keywords

Distribution Function Characteristic Function Bounded Function Satisfy Condition Uniqueness Theorem
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## Notes

- §1.For the basic theory of Stieltjes integrals see, for instance, Burkill and Burkill [1], Ch.6, and Widder [1], Ch.I.Google Scholar
- §2.As Zygmund has remarked, the essence of Theorems 3 and 4 is a classical result of the calculus of probability, in a form strengthened by Cramér. See Zygmund [1], Vol.11, Ch.XVI, Th.(4.24), p.262. See also Cramér [2], Ch.10. Bochner has a generalization to E
_{k}, see Th.3.2.1 of his book [5], p.56. For Helly’s theorem used in the proof of Theorem 4, see, for instance, Widder [1], Ch.1, §16, Th.16.2.Google Scholar - §3.Theorems 6 and 7 are due to Bochner, see [1], Th.23. He refers to previous work by F. Bernstein and M. Mathias. The generalization,-without the assumption of continuity, is due to F. Riesz [1], who uses for the proof, however, his theorem on the representation of positive linear functionals, which is
*not*used here in the proof of Theorem 5. For the Helly-Bray theorem used, see, for instance, Widder [1], p.31, Th.16.4. It is not necessarily true when the interval of integration is infinite, as Widder makes clear, hence the introduction of the kernel K_{R}(x). Carleman [1], p.98, gives a proof of Bochner’s theorem*using*the Poisson integral representation of functions which are positive and harmonic in a half-plane. A proof of the latter (see, for instance, Verblunsky [1]) can be obtained by using Herglotz’s theorem [1] on the representation of positive, harmonic functions in a circle (which is stated, for instance, in Stone [1], p.571), or more directly, as has been done by Loomis and Widder [1] using the theorems of Helly, and of Helly-Bray. It should be remarked, however, that all these representation theorems are, more or less, of the same order of difficulty as Bochner’s theorem, or Stone’s spectral theorem [1], p.331, as was early recognized by F. Riesz. Apropos Corollary (3.27), see Cramér [1]. For Theorem 9 see Bochner [3], p.329. Bochner has also a generalization to E_{k}, [5], Theorem 3.2.3, p.58. For a generalization to distributions, see Schwartz [1], Vol.11, p.132, Th.XVIII; Schwartz makes a reference to Weil [1], p.122.Google Scholar - §4.Lemma 1 is due to H.A. Schwarz. It is quoted by G. Cantor,
*J. für Math*. 72 (1870), 141; and is given by Schwarz himself in his*Ges. Abhandlungen*, II (1890), 341–343, with a reference to Cantor’s quotation. Prof. Raghavan Narasimhan has remarked that a rearrangement of Schwarz’s argument is better adapted to generalization. “If f is real-valued, and continuous on (a,b), and \(\lim_{h\rightarrow 0} sup (\Delta _h 2)(\textup{x})=0\), for all x∈ (a,b), where \(\Delta _\textup{h} 2\textup{f}(\textup{x})=\textup{h} {-2} \{\textup{f}(\textup{x+h})+\textup{f}(\textup{x-h})-2\textup{f}(\textup{x})\}\), then f is linear. To prove this, it is sufficient to prove that if \(\lim_{\textup{h}\rightarrow 0}\textup<Superscript>p</Superscript>\Delta _\textup{h} 2\textup{f}\geq 0\), then f is convex (i.e. if ℓ(x) = cx+d, and f(α)≤ℓ(α), f(β)≤ℓ(β), where a < α < β < b, then f (x) ≤ℓ(x) for α ≤ x≤ β), since one can argue similarly with \(\lim_{\textup{h}\rightarrow 0}\textup<Subscript>f</Subscript>\Delta _\textup{h} 2(\textup{-f})\). \(\textup{Since}\Delta _\textup{h} 2l=0\) for any linear function ℓ, it is enough to prove that if \(\lim_{h\rightarrow 0}\textup<Superscript>p</Superscript>\Delta _\textup{h} 2\textup{f}\geq 0\), and f(α) ≤ 0, f(β) ≤ 0, then f(x)≤0 on [α, β], i.e. that f has no maximum on (α, β). Replacing f(x) by f(x) + εx, ε> 0, one has only to show that if \(\lim_{h\rightarrow 0}\textup<Superscript>p</Superscript>\Delta _\textup{h} 2\textup{f}> 0\), on (a,b), then f has no local maximum on (a,b). But this is obvious, for if x_{0}is a local maximum, then \(\Delta _\textup{h} 2\textup{f}(x_0)\leq 0\) for h small enough, \(\textup{since f}(\textup{x}_0)\geq \textup{f}(\textup{x}_0+\textup{h}), \textup{f}(\textup{x}_0)\geq \textup{f}(\textup{x}_0-\textup{h})\).” Cf. Narasimhan [2], p.21–25. Theorem 10 is due to A.C. Offord [1], and is the integral analogue of Cantor’s fundamental theorem on the uniqueness of trigonometric series, which asserts that if a trigonometric series converges everywhere to zero, it vanishes identically; all its coefficients are zero. See, for instance, Zygmund [1], Vol.1, Ch.IX, p.326. Offord also proved [2] a stronger theorem in which the hypothesis of convergence of the integral in (4.1) is replaced by (*C*,1) summability. Offord shows that the stronger theorem is a “best possible”, in the sense that even one exceptional point cannot be permitted, and (*C*,1) summability cannot be relaxed to (*C*,1+ε) summability for any ε > 0. Zygmund’s proof [1], Vol. II, Ch.XVI, §10, of Offord’s first theorem is based on an equicon-vergence theorem for trigonometric integrals and series which he treats in Vol.11, Ch.XVI, §9, and on results from Riemann’s theory of trigonometric series which he treats in Vol.I, Ch.IX. For the use of equiconvergence theorems in analytic number theory, see, for instance, the author’s book [3], Ch.VIII. A generalization of Offord’s stronger theorem to several variables would be of interest, though perhaps not easy. An equiconvergence theorem for trigonometric integrals in two variables has been given by H. Keller [2]. Functions of bounded variation in two variables come into play, and it is a moot question whether the notion of Vitali variation could be replaced by that of Frechet, as Morse [1] and Transue did in another context.Google Scholar

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