Product Summability and Approximation of Functions in Lipschitz classes

Abstract

This paper contributes to the theory of approximation by establishing a relation on the degree of approximation for functions belongs to Lipschitz function. Utilizing the product summability method (C, 1) (E,q), we investigate the behavior of the Fourier series associated with such functions. The findings offer deeper insight into the convergence dynamics and approximation precision of product summation methods contained in the Lipschitz framework. By demonstrating how these methods surpass classical summability in handling non-smooth functions, this work highlights their robustness and potential in harmonic analysis. The results not only reinforce the utility of product summability in Fourier approximation but also contribute to a broader theoretical understanding of summability methods in modern mathematical analysis.