Analysis of Euler-Banach operator to approximate the function using its Fourier series

The Fourier series, known for expressing functions as sums of sines and cosines, can be refined in various ways to improve convergence and achieve more accurate signal approximation.Utilizing a product transform increases the convergence rate, resulting in a closer representation of the original signal.In this work, we introduce the notion of Euler-Banach operator to approximate functions in the Lebesgue class through the Fourier series and its conjugate series and also to establish two approximation theorems using our proposed summation operator.