ENHANCING FOURIER SERIES: POLYNOMIAL EXPANSION FOR APERIODIC FUNCTIONS ON LIMITED INTERVALS

Fourier series have long been a cornerstone of mathematical analysis, particularly for periodic functions defined on intervals [-π, π] or [0, 2π], where the period 'T' plays a central role. However, when dealing with aperiodic functions defined on limited intervals like [a, b], the conventional approach involves expanding them into periodic functions within the same domain. Commonly, this is achieved through half-range sine or cosine expansions, but this approach often results in periodic functions that are discontinuous or lack differentiability, causing slow convergence of the corresponding Fourier series. To address this issue and accelerate convergence, polynomial interpolation methods have been devised, modifying the original function using Lanczos polynomials to attain the desired smoothness. Additionally, alternative forms of Fourier series, as explored in previous works, introduce extra sine or cosine terms to the series, subsequently determining these extra terms to enhance convergence based on the desired smoothness. In this paper, we extend the discussion beyond periodic functions by considering Fourier series expansion for functions defined on a general interval [a, b]. We introduce polynomial expansions that expand these functions to a larger domain, thereby achieving the desired level of smoothness and enhancing the convergence of Fourier series.