Mathematics and Computer Science: Research Updates Vol. 12

Product summability methods provide a useful framework for studying the approximation of Fourier series when ordinary convergence does not fully capture the behaviour of functions with limited smoothness. This study examines the degree of approximation of periodic functions belonging to the Lipschitz class by applying the product mean obtained from the Cesàro and Euler summability methods to the associated Fourier series. The work is placed within summability theory and harmonic analysis, where the control of approximation error is central to understanding the convergence of transformed trigonometric series. The manuscript first reviews the relevant notions of degree of approximation, Lipschitz continuity, Cesàro summability, Euler summability and product means, and then establishes the principal theorem for functions that are Lebesgue integrable on the stated interval and periodic with period 2π. The theorem gives an estimate for the approximation error in relation to the smoothness parameter of the Lipschitz class and the order of the Fourier approximation. The analysis indicates that the combined Cesàro–Euler product mean provides a structured procedure for obtaining approximation estimates for functions whose regularity is controlled but not necessarily differentiable. The results contribute to the study of summability-based Fourier approximation by clarifying how a product method can be used in the Lipschitz setting. The discussion further notes the relevance of such approximation techniques to areas in which Fourier representations are used, including signal analysis, numerical methods and computational modelling. Overall, the study presents a mathematically focused treatment of product summability as a tool for estimating the approximation of Lipschitz functions through Fourier series, without extending the conclusion beyond the established theorem.