Approximation of Periodic Functions by Fejer Sum and De La Vallee Poussin Sums

In this paper, we establish several results concerning the approximation of periodic functions by Fejér and de la Vallée Poussin means in Lebesgue spaces L₂πᵖ. The obtained estimates are expressed in terms of function for L₂ and the second-order modulus of continuity. The approximation of periodic functions by trigonometric polynomials plays a central role in Fourier analysis. Among the classical summation methods, Fejér sums and de la Vallée-Poussin sums provide powerful tools for improving the convergence behavior of Fourier series. In this work, we investigate the approximation of -periodic functions in spaces L₂by these two summation methods. Special attention is given to the relationship between the smoothness of the function, measured via the second-order modulus of continuity, and the rate of approximation. Our results contribute to a clearer understanding of how summability methods refine Fourier approximation and provide effective tools for both theoretical and applied analysis.