The Wiener algebra of absolutely convergent Fourier integrals: an overview
In this nice survey of 68 pages written by three experts in the area, the main results in the topic of representation of non-periodic functions by absolutely convergent Fourier integral are collected, classified, discussed and applied. The history of the topic has its origin in the classical results of S. N. Bernstein and of A. Zygmund, who proved that if a periodic function \(f\) is from the Lip\(\alpha\) class with \(\alpha >1/2\), then for the absolutely convergence of its Fourier series, a sufficient condition is \(\int_{0}^{1}\frac{\omega(f, t)}{t^{3/2}}dt <+\infty\) (S. N. Bernstein's result), and if in addition \(f\) is of bounded variation then a sufficient condition is \(\int_{0}^{1}\frac{\sqrt{\omega(f, t)}}{t}d t < +\infty\) (A. Zygmund's result). Here \(\omega(f, t)\) denotes the modulus of continuity of \(f\). Extensions to the non-periodic case were started in the \(20-40\)s of the twentieth century, with E. C. Titchmarsh's and N. Wiener's results, then becoming a well-studied area of research until our days. This survey consists of 13 sections and an updated list of references. Sections 1, 2 and 3 contains the Introduction, Notation and Definitions and Preliminaries, respectively. Section 4 presents general properties of the Wiener algebras. Sufficient conditions for a function \(f:\mathbb{R}\to \mathbb{C}\) to belong to the Wiener algebra \(W_{0}(\mathbb{R})=\{f ; f(x)=\int_{\mathbb{R}}\mathrm e^{-\mathrm i x y}g(y)d y , g\in L_{1}(\mathbb{R}^{n})\}\) are presented in the next sections 6-12. From a very long list, from each section we selected the following results. Theorem 1. (Tests in one-dimensional case) (i) Let \(f\in L_{p}(\mathbb{R})\bigcap C_{0}(\mathbb{R})\), \(1<p\leq 2\). If \(\int_{\mathbb{R}}|f(x+h)-f(x)|^{p}dx\leq Ch^{\alpha p}\), for all \(0<h\leq 1\), where \(1/p < \alpha \leq 1\), then \(f\in W_{0}(\mathbb{R})\); (ii) If \(f\) satisfies the condition \(|f(x)-f(y)|\leq C\frac{|x-y|^{\lambda}}{(1+|x|)^{\lambda}(1+|y|)^{\lambda}}\), for all \(x, y\in \overline{\mathbb{R}}\), with \(\lambda > 1/2\) and \(f(\infty)=0\), then \(f\in W_{0}(\mathbb{R})\); (iii) Let \(f\in C_{0}(\mathbb{R})\) be twice continuously differentiable on \(\mathbb{R}\) such that \(\int_{0}^{\infty}x|f^{\prime \prime}(x)+f^{\prime \prime}(-x)|dx <\infty\) and there exists \(\sigma >0\) with \(\int_{0}^{\infty}x^{1+\sigma}\left |\left (\frac{f(x)-f(-x)}{x^{\sigma}}\right )^{\prime \prime }\right | dx <\infty\). Then \(f\in W_{0}(\mathbb{R})\). Theorem 2. (Tests in terms of integrability of derivatives) (i) Let \(f\in L_{2}(\mathbb{R}^{n})\) and \((-\Delta)^{\alpha/2}(f)\in L_{e}(\mathbb{R}^{n})\) with \(\alpha \in (n/2, +\infty)\) not necessarily integer. Then \(f\in W_{0}(\mathbb{R}^{n})\) (here \(\Delta\) denotes the Laplacian); (ii)If \(f\in L_{1}(\mathbb{R}^{n})\) and \(f\) has mixed distributional derivatives \(D^{j}(f)\in L_{p}(\mathbb{R}^{n})\) for all \(j\in [0, 1]^{n}\), \(j\not=0\), where \(1<p\leq 2\), then \(f\in W_{0}(\mathbb{R}^{n})\). Theorem 3. (Tests in terms of finite difference) (i) (Bernstein-type result) Let \(f\in C(\mathbb{R}^{n})\) satisfying the condition \(\int_{0}^{\infty}\sup_{0<|h|<t}\|\Delta^{m}_{h}(f)\|_{L_{2}}\frac{dt}{t^{1+n/2}}<\infty\), with \(m>n/2\). Then \(f\) is representable as the sum of the Fourier transform of a function in \(L_{1}(\mathbb{R}^{n})\) and a polynomial of degree \(\leq m-1\); (ii) (Zygmund-type result) Let \(f\in C^{q}(\mathbb{R}^{n})\) be compactly supported, where \(q=[(n-1)/2]\). If the partial derivative \(D_{q, j}(f):=\frac{\partial^{q} f}{\partial x_{j}^{q}}\) has , as a function of \(x_{j}\), a bounded, with respect to the other variables, number of points separating the intervals of convexity for each \(1\leq j\leq n\), then the condition \(\int_{0}^{1}\omega(t)t^{q-(n+1)/2} dt <\infty\), where \(\omega(t)=\max_{j=1,..., n}\{\omega(D_{q, j}(f) ;t)\}\), yields \(f\in W_{0}(\mathbb{R}^{n})\). Theorem 4. (Tests for radial and quasi-radial functions) (i) If \(g\in BV_{k+1}\) (a subspace of functions with derivative of order \(k+1\) with bounded variation, see Definition 3.1 in the paper), where \(k=[(k+1)/2]\), then \(g(|x|^{\lambda})\in W_{0}(\mathbb{R}^{n})\); (ii) Let \(f(x)=f_{0}(x)\), where \(N>(n-1)/2\) and \(f\) is differentiable up the order \(N\), with \(g\) and \({\mathcal{D}}^{N}(g)\) nonincreasing and tending to zero as \(r\to \infty\) (here \({\mathcal{D}}^{1}g(r)=-(1/r)g^{\prime}(r)\), \({\mathcal{D}}^{N}(g)={\mathcal{D}}^{1}({\mathcal{D}}^{N-1}(g))\)). Denote \(A_{N}:=\int_{0}^{\infty}r^{2N-1}|{\mathcal{D}}^{N}f_{0}(r)|dr<\infty\). Then \(f\in W_{0}(\mathbb{R}^{n})\) and \(\|f\|_{W_{0}}\leq C A_{n}\) with \(C\) independent of \(f\). Theorem 5. (Tests with fractional derivatives) (i) Let \(f, \mathbb{D}^{\alpha}(f)\in L_{2}(\mathbb{R})\), where \(\mathbb{D}^{\alpha}(f)\) denotes the Riesz fractional derivative and \(\alpha > 1/2\). Then \(\|f\|_{W_{0}}\leq C \|f\|_{L_{2}}^{1-1/(2\alpha)}\cdot \|\mathbb{D}^{\alpha}(f)\|_{L_{2}}^{1/(2\alpha)}\). (ii) Let \(f_{0}\in BV_{k+1}(\mathbb{R}_{+})\), where \(\alpha > (n-1)/2\). Then \(f(x)=f_{0}(|x|)\in W_{0}(\mathbb{R}^{n})\). Theorem 6. (Test for functions with singular derivatives) Let \(f\in C^{N}(\mathbb{R}^{n}\setminus \{0\})\), \(N=1+[n/2]\) have compact support and let there exist constants \(C>0\) and \(\delta >0\) such that \(|D^{j}(f)(x)|\leq C|x|^{\delta -|j|}, x\in \mathbb{R}^{n}\setminus \{0\}\), for all \(j\) with \(|j|=0, 1, ..., N\). Then \(f\in W_{0}(\mathbb{R}^{n})\). Theorem 7. (Tests for positive definite functions) (i) \(f:\mathbb{R}\to \mathbb{C}\) satisfying \(f(t)=0\) for \(|t|\geq \sigma >0\) belongs to \(W_{0}^{+}(\mathbb{R})=W_{0}(\mathbb{R})\bigcap \{\text{ the set of positive definite functions } \}\), if and only if there exists \(g\in L_{2}(\mathbb{R})\), \(g(t)=0\) for \(|t|\geq \sigma/2\), such that for any \(t\in \mathbb{R}\), \(f(t)=\int_{-\infty}^{+\infty}g(y)\overline{g(t+y)}dy\); (ii) Each even, convex and monotone decreasing to zero function on \([0, \infty)\) belongs to \(W_{0}^{+}(\mathbb{R})\). Theorem 8. (Test for convex and concave functions) (Zygmund) If an odd function \(f\) is concave on \(\mathbb{R}_{+}\), then \(f\in W_{0}(\mathbb{R})\) locally (that is can be extended from any interval in such a way that the extension will belong to \(W_{0}(\mathbb{R})\)) if and only if the integral \(\int_{0}^{1}\frac{f(t)}{t}dt\) converges. Also, for functions in the intermediate classes \(V^{*}\) between that of functions with bounded variation and the class of functions that are linear combination of convex functions each, sufficient conditions to belong to \(W_{0}(\mathbb{R})\) are presented. Sections 12 ends with some necessary and sufficient conditions for the summability of Fourier series. In the last section 13, two very useful tables, one containing Fourier transforms for various integrable functions and the other one containing positive definite functions depending on parameters, are presented.
scientific article; zbMATH DE number 6112460
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