On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

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


We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $$$2\pi$$$-periodic functions $$$f$$$ with absolutely convergent Fourier series, for $$$f$$$ to belong to the generalized Lipschitz classes $$$H^{\omega, \alpha}_{\mathbb{C}}$$$, and to have the fractional derivative of order $$$\alpha$$$ ($$$0 < \alpha < 1$$$).


Fourier series; $$$\alpha$$$-th modulus of continuity; generalized Lipschitz classes


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

Moricz F. "Absolutely convergent Fourier series and function classes", J. Math. Anal. Appl., 2006; 324: pp. 1168-1177. doi:10.1016/j.jmaa.2005.12.051

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

DOI: https://doi.org/10.15421/241613



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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991