Some properties of Fourier integrals

2011/08/14 by A. F. Grishin, Grishin, A. F., Maksym Skoryk +1
Mathematics · #42A38 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1108.2890

openalex publication_date 2011/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F(Rⁿ) be the algebra of Fourier transforms of functions from L₁(Rⁿ), K(Rⁿ) be the algebra of Fourier transforms of bounded complex Borel measures in Rⁿ and W be Wiener algebra of continuous 2pi-periodic functions with absolutely convergent Fourier series. New properties of functions from these algebras are obtained. Some conditions which determine membership of f in F(R) are given. For many elementary functions f the problem of belonging f to F(R) can be resolved easily using these conditions. We prove that the Hilbert operator is a bijective isometric operator in the Banach spaces W₀, F(R), K(R)-A₁ (A₁ is the one-dimension space of constant functions). We also consider the classes M_k, which are similar to the Bochner classes F_k, and obtain integral representation of the Carleman transform of measures of M_k by integrals of some specific form.

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