An exact constant on the estimation of the approximation of classes of periodic functions of two variables by Ceśaro means

Abstract

In the present work, we study problem related to the approximation of continuous $2\pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2\pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f \in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others. The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.