Exploring Chebyshev Polynomial Approximations: Error Estimates for Functions of Bounded Variation

TL;DR

This paper investigates Chebyshev polynomial approximations for functions of bounded variation, providing optimal error estimates and validating them through numerical experiments, with implications for signal processing and machine learning.

Contribution

It introduces two optimal error bounds for Chebyshev approximations of bounded variation functions and demonstrates their effectiveness through numerical validation.

Findings

Optimal error estimates for Chebyshev approximations of bounded variation functions.

Numerical experiments confirm the theoretical error bounds.

Potential applications in machine learning and signal processing.

Abstract

Approximation theory plays a central role in numerical analysis, undergoing continuous evolution through a spectrum of methodologies. Notably, Lebesgue, Weierstrass, Fourier, and Chebyshev approximations stand out among these methods. However, each technique possesses inherent limitations, underscoring the critical importance of selecting an appropriate approximation method tailored to specific problem domains. This article delves into the utilization of Chebyshev polynomials at Chebyshev nodes for approximation. For sufficiently smooth functions, the partial sum of Chebyshev series expansion offers optimal polynomial approximation, rendering it a preferred choice in various applications such as digital signal processing and graph filters due to its computational efficiency. In this article, we focus on functions of bounded variation, which find numerous applications across mathematical physics, hyperbolic conservations, and optimization. We present two optimal error estimations associated with Chebyshev polynomial approximations tailored for such functions. To validate our theoretical assertions, we conduct numerical experiments. Additionally, we delineate promising future avenues aligned with this research, particularly within the realms of machine learning and related fields.