Knowledge, Innovation and Technology in Scientific Research Vol. 1

Approximation theory provides a framework for assessing how effectively Fourier series represent periodic functions, while summability methods can improve convergence when ordinary partial sums are inadequate. This study examines the degree of approximation of periodic functions by a generalised Cesàro–Euler product mean of Fourier series. The analysis is developed for periodic functions that are Lebesgue integrable and belong to a Lipschitz class. The proposed framework combines the averaging property of Cesàro means with the smoothing property of Euler means and is formulated through the corresponding product summability kernel. Using the stated definitions, auxiliary lemmas, and the Lipschitz condition, a main theorem is established to characterise the order of approximation under the specified parameter ranges. The derived result shows how the approximation order depends on the smoothness parameter and the order associated with the generalised product mean. Particular parameter choices also recover relevant special cases of the product method, including the Cesàro mean and a Cesàro–Euler product form. Within the assumptions considered, the analysis supports the use of the generalised Cesàro–Euler product mean as a systematic approach for studying convergence and approximation of periodic functions. The findings extend the discussion of product summability in Fourier approximation and provide a theoretical basis for further investigation of related function classes and summability procedures.