Fourier transforms in the complex domain. (English) Zbl 0011.01601

Colloquium Publications. American Mathematical Society 19. New York: Am. Math. Soc. viii, 184 S. (1934).
As stated in the preface, the book covers a great variety of topics, but is unified by the central idea of application of the Fourier transform in the complex domain. The variety is great indeed. We shall enumerate the main characteristic theorems treated in the book, most of which have been published previously by the authors in one form or another, or are known theorems with new proofs and appearing in new connections.
(1) A new proof of Carleman’s theorem on quasi-analytic functions, employing the remarkable theorem that a function of class \(L_2\) in \(\-\infty < x < \infty\) vanishes on a half-line if and only if its Plancherel transform \(\varphi(x)\) makes the integral \[ \int_{-\infty}^\infty \frac{\vert\log \vert\varphi(x)\vert \vert}{1+x^2}\, dx \] finite.
(2) A new proof of Szász’s theorem that the set of functions \(\{x^{\lambda_n}\}\), \(\operatorname{Re}(\lambda_n) > -\tfrac12\), will be closed \(L_2\) over \((0, 1)\) when, and only when, \[ \sum_{n=1}^\infty \frac{1+2\operatorname{Re}(\lambda_n)}{1 + \vert\lambda_n\vert^2} \] diverges.
(3) Solution of the equation \[ f(x) = \int_0^\infty K(x-y)f(y) \,dy \] (the lower limit of the integral 0, not \(-\infty\)!).
(4) Theorems on the distribution of zeros for entire functions \(F(z)\) of exponential type which, if specifically applied to functions of the form \(F(z) = \int_b^a e^{ixx} f(x)\,dx\) give (4) fine theorems about the closure of sets of functions \(\{e^{i\lambda_nx}\}\) and \(\{e^{\pm i\lambda_nx}\}\) over a finite interval. For instance, if \(\vert \lambda_n - n\vert \le L < \frac1{\pi^2}\), \(n = 0, \pm 1, \ldots\), then the set of functions \(\{e^{ i\lambda_nx}\}\) is closed \(L_2\) over \((-\pi, \pi)\), and admits a closed normal biorthogonal set \(\{h_n(x)\}\) with interesting properties.
(5) A novel gap-theorem on almost-periodic functions, and lastly
(6) Wiener’s differential space. Since the present exposition of this subject differs somewhat from previous ones we shall give a terse exposition of it.
Let each variable \(\alpha_n\), \(n = 0, \pm 1, \pm 2, \ldots\) represent the interval \(0\le \alpha_n \le 1\) with the usual Lebesgue measure, and let the space of points \(\alpha = (\ldots, \alpha_{-n}, \ldots, \alpha_0, \ldots, \alpha_n, \ldots)\) have the modern measure of an infinitely dimensional torus, and let \(a_n\) be \((-\log \alpha_{2n-1})^{1/2} e^{2\pi i\alpha_{2n}}\). Then the series \[ \psi(x, \alpha) \sim xa_0 + \sum_1^\infty \frac{a_n e^{2\pi inx}}{in} + \sum_1^\infty \frac{a_n e^{-2\pi inx}}{-in} \] is for almost all \(\alpha\) the Fourier series of a continuous function in \(x\). In fact, for almost all \(\alpha\) \[ \varlimsup_{\varepsilon\to 0} \vert \psi(x+s, \alpha) - \psi(x, \alpha)\vert \varepsilon^{-\lambda} \] is 0 uniformly in all \(x\), whenever \(\lambda < \frac12\) (; but, whenever \(\lambda > \frac12\) for almost all \(\alpha\) it is infinite for all \(x)\). Let \(F(x) = \sum_{-\infty}^\infty f_n e^{2\pi inx}\) be a function of \(L_2\). For almost all \(\alpha\), the integral \(\int_0^1 F(x) \,d\psi(x, \alpha)\) exists as the sum \({\sum'}_{-\infty}^\infty f_na_n\), and if \(F_1, \ldots F_k\) are orthonormal then for an arbitrary non-negative measurable function \(\Phi(z_1, \ldots, z_k)\) of the complex variables \(z_1, \ldots, z_k\), \[ \int_0^1 \Phi \left\{ \int_0^1 F_1(x)\,d\psi(x,\alpha), \ldots, \int_0^1 F_k(x)\,d\psi(x,\alpha)\right\} \, d\alpha = \frac1{\pi^{k/2}} \int_{-\infty}^\infty \cdots \int \Phi(z_1, \ldots, z_k) e^{-\vert z_1\vert^2 - \cdots - \vert z_k\vert^2} \,dz. \tag{*} \]
Furthermore, if \(T^t\), \(-\infty < t < \infty\), is a continuous group of unitary transformations in \(L_2\), for which \[ \lim_{t\to\infty} \int_0^1 \overline{F_p(x)} T^tF_q(x) \,dx = 0, \] then, for almost all \(\alpha\), \[ \lim_{A\to\infty} \frac1{A} \int_0^A \Phi\left\{ \int_0^1 T^t F_1(x)\,d\psi(x,\alpha),\ldots, \int_0^1 T^t F_k(x)\,d\psi(x,\alpha)\right\}\,dt \] is equal to the common value of (*); (Birkhoff’s ergodic theorem). Special application in the case \(T^tF(x) = F(x+t)\).


42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces