On the Degree of Approximation of Periodical Functions Using Generalized Cesaro-Euler Product Means of Fourier Series
Priyadarshani Behera
Department of Mathematics, Driems University, Cuttack, Odisha, Pin-754022, India.
Ranjan Kumar Jati *
Department of Mathematics, Driems University, Tangi, Cuttack-754022, India.
Nirmal Chandra Sahu
Department of Mathematics, Driems University, Tangi, Cuttack-754022, India.
*Author to whom correspondence should be addressed.
Abstract
Approximation theory and Fourier analysis have long been important areas of mathematical research due to their wide applications in harmonic analysis, numerical computation, and signal processing. In recent years, generalized summability methods, particularly combinations of Cesàro and Euler means, have attracted considerable attention for their ability to improve the convergence and approximation behavior of Fourier series for periodic functions. The idea of rate of approximation has been widely applied in various mathematical fields and approximation theory to measure accuracy. The present manuscript contributes to the field of approximation theory and Fourier analysis by investigating the degree of approximation of periodic functions using generalized Cesàro–Euler product means. The work is mathematically relevant because it combines Cesàro and Euler techniques into a generalized framework that may be useful in harmonic analysis, numerical approximation, and signal processing applications. The article also opens possibilities for future research on generalized summability methods and multidimensional Fourier approximations.
Keywords: Order of approximation, Lipschitz functions, Cesaro summability, Euler summability, product mean, lebesque integral.