Exploring Chebyshev polynomial approximations: Error estimates for functions of bounded variation

Approximation theory plays a central role in numerical analysis, evolving through a variety of methodologies, with significant contributions from Lebesgue, Weierstrass, Fourier, and Chebyshev approximations. For sufficiently smooth functions, the partial sum of Chebyshev series expansion provides optimal polynomial approximation, making it a preferred choice in many applications. However, existing literature predominantly focuses on Chebyshev interpolation, which requires exact Chebyshev series coefficients. The computation of these exact coefficients is challenging and often impractical for numerical algorithms, limiting their practical utility. Additionally, traditional approaches typically involve polynomials on fixed intervals where the basis functions of the series are defined. In this article, we have generalized Chebyshev polynomial approximation to a broader domain and presented two optimal error estimations for functions of bounded variation, using approximated Chebyshev series coefficients. This aspect is notably absent in current literature. To support our theoretical findings, we conducted numerical experiments and proposed future research directions, particularly in the fields of machine learning and related areas.