Approximation by double matrix means of bivariate functions belonging to weighted Lipschitz and Zygmund classes

Abstract

This study focuses on the rate of uniform approximation for continuous functions ‐periodic in each variable, using double matrix means of the rectangular partial sums of double Fourier series. The importance of the weighted integral modulus of symmetric smoothness lies in its ability to capture the local behavior of a function. So we obtain the results in terms of the weighted integral modulus of symmetric smoothness. In addition, we introduce the concept of the weighted Lipschitz and Zygmund classes as an extension of the existing Lipschitz classes and and Zygmund classes and , respectively. These weighted classes allow us a more detailed analysis of the approximation rates of functions by assigning the weights to different regions of the domain of function. Further, we prove two theorems about the degree (error) of approximation of functions belonging to these classes using double matrix means. We also discuss some corollaries from our results.