Knowledge, Innovation and Technology in Scientific Research Vol. 1

Function approximation by Fourier series is an important topic in mathematical analysis because the quality of an approximation depends on both the smoothness of the function and the summability procedure applied to its Fourier series. Classical partial sums may exhibit convergence limitations and oscillatory behaviour, motivating the use of product summability methods. This study examines the degree of approximation of periodic, Lebesgue-integrable functions belonging to the Lipschitz class Lip(α,p) by the (C,1)(E,1) product mean of their Fourier series. The analysis uses the manuscript’s definitions of the degree of approximation, Lipschitz classes, Cesàro summability, Euler summability, Fourier series, and Big O notation. The principal result is obtained by applying the stated product-mean framework and specialising the relevant conditions to the (C,1)(E,1) case. The resulting estimate describes the approximation error in relation to the smoothness and integrability conditions of the function. The accompanying corollaries indicate that, under the stated assumptions, the approximation error decreases as the order of the approximating polynomial increases. The discussion further shows that combining Cesàro and Euler averaging can provide smoother convergence and improved error control compared with using the individual summability procedures separately. Overall, the study establishes a product-summability formulation for Fourier approximation within the specified Lipschitz class and provides a basis for further analytical investigation of related summability methods.