In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

The Wiener algebra of absolutely convergent Fourier integrals: an overview

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of ${\mathcal{B}}(\ell^{q}_{w})$, the algebra of all bounded linear operators on the weighted sequence space $\ell^{q}_{w}$, for any 1≤q<∞ and any discrete Muckenhoupt Aq-weight w.

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Wiener’s lemma for infinite matrices

Summation of Multiple Fourier Series by Spherical Means

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of ${\mathcal B}(\ell^q_w)$, the algebra of all bounded linear operators on the weight sequence space $\ell^q_w$, for any $1\le q<\infty$ and any discrete Muckenhoupt $A_q$-weight $w$.

Wiener's Lemma for Infinite Matrices II

On some singular Fourier multipliers = ある特異なフーリエ掛け算作用素について

Beurling's algebra $A^*=\{f:\sum_{k=0}^{\infty} \sup_{k\le |m|} |\hat f (m)| < \infty \}$ is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener's algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with $[-\pi,\pi],$ and its dual space is indicated. Analogs of Herz's and Wiener-Ditkin's theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.

The banach algebra A and its properties

We compare the Fourier integral of a function of bounded variation and the corresponding trigonometric series, generated by that function, in the multidimensional case Several known notions of bounded variation are used and a new one is introduced The obtained results are applied to integrability of multidimensional trigonometric series

An Interplay of Multidimensional Variations in Fourier Analysis

New sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension $d\ge2.$

R. Trigub

Papers

The Wiener algebra of absolutely convergent Fourier integrals: an overview

The banach algebra A and its properties

An Interplay of Multidimensional Variations in Fourier Analysis

On absolute convergence of Fourier integrals