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

Abstract

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 $$$\alpha$$$ ($$$0 < \alpha < 1$$$).