The Properties of Ces\'aro General Fourier Sums of Functions with Derivatives of Lipschitz Class Functions

Abstract

In this paper, we investigate the Ces\'aro means of Fourier series with respect to general orthonormal systems (ONS), when the function \( f \) belongs to a certain differentiable class of functions. It is well known that the membership of a function \( f \not\equiv 0 \) in a differentiable class does not, in general, guarantee the summability of its Fourier series with respect to an arbitrary ONS. Therefore, in order for the Fourier series with respect to a given ONS to be summable, one must impose additional conditions on the system functions \( \{\varphi_n\} \). The main objective of this work is to determine such conditions on the functions \( \varphi_n \) of the ONS under which the Ces\'aro means of the Fourier series of any function whose derivative belongs to the Lipschitz class \( \mathrm{Lip}_1 \) are uniformly bounded. The results obtained are sharp in the sense that the conditions cannot be essentially weakened.