Absolute convergence of Fourier integrals and Lipschitz classes defined with differences of fractional order

B.I. Peleshenko (Dnipropetrovsk State Agrarian and Economic University)
T.N. Semirenko (Dnipropetrovsk State Agrarian and Economic University)

Abstract


The necessary and sufficient conditions in terms of Fourier transforms $$$\hat{f}$$$ of functions $$$f\in L^1(\mathbb{R})$$$ are obtained for $$$f$$$ to belong to the Lipschitz classes $$$H_C^{\omega, \alpha}(\mathbb{R})$$$ and $$$h_C^{\omega, \alpha}(\mathbb{R})$$$, defined by differences of fractional order.

Keywords


Fourier transform; Fourier integral; modulus of continuity; Lipschitz classes

References


Stein I., Weiss G. Introduction to Fourier analysis on Euclidean spaces, Mir, Moscow, 1974; 333 p. (in Russian)

Moritz F. "Absolutely convergent Fourier integrals and classical function spaces", Arch. Math., 2008; 91: pp. 49-62.

Peleshenko B.I. "Absolute convergence of Fourier integrals and Lipschitz classes", Dnipr. Univ. Math. Bull., 2011; 16: pp. 102-108. (in Russian)


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Copyright (c) 2013 B.I. Peleshenko, T.N. Semirenko

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ISSN (Online): 2518-7996
ISSN (Print): 2312-9557