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

S Akansha

TL;DR

This work develops generalized decay bounds for Chebyshev series coefficients of functions of bounded variation on general intervals, not just $[-1,1]$, and uses these to obtain sharp $L^1$-error estimates for truncated Chebyshev approximations with approximated coefficients. By extending two established decay results (Majidian and Xiang) to $[a,b]$, the authors derive explicit bounds that remain valid under BV regularity and when exact coefficients are unavailable. Theoretical results are complemented by numerical experiments that verify the improved bounds and highlight practical benefits for spectral methods and related applications. The findings have potential impact in areas such as digital signal processing, graph signal processing, and machine learning, where efficient and reliable Chebyshev approximations on general domains are needed.

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.

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

TL;DR

This work develops generalized decay bounds for Chebyshev series coefficients of functions of bounded variation on general intervals, not just , and uses these to obtain sharp -error estimates for truncated Chebyshev approximations with approximated coefficients. By extending two established decay results (Majidian and Xiang) to , the authors derive explicit bounds that remain valid under BV regularity and when exact coefficients are unavailable. Theoretical results are complemented by numerical experiments that verify the improved bounds and highlight practical benefits for spectral methods and related applications. The findings have potential impact in areas such as digital signal processing, graph signal processing, and machine learning, where efficient and reliable Chebyshev approximations on general domains are needed.

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.
Paper Structure (10 sections, 7 theorems, 52 equations, 1 figure)

This paper contains 10 sections, 7 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chebyshev Series Expansion
  4. Gauss Chebyshev quadrature formula
  5. Notations
  6. Decay bounds for Chebyshev Coefficients
  7. Error Estimate for Chebyshev Approximation
  8. Numerical Comparison
  9. Conclusion and Future Directions
  10. Conclusion and Future Directions

Key Result

Lemma 2.1

For a given positive integer $n$, we have for any integer $k$ such that $0\le k< 2n$.

Figures (1)

  • Figure 1: (a) Depicts the comparison between the decay bounds of $|c_j|$, for $j=2,3,\ldots, 30$ derived in Theorem \ref{['thm:errestimate']} (blue line) and Theorem \ref{['thm:decayc_jinV_k']} (red line). (b) Depicts the comparison between the error estimates we obtain in Theorem \ref{['thm:errestimate']} (blue line) and Theorem \ref{['thm:errestxiang18']} (red line) for $d=n-l$, where $n=200$ and $l=2^j,j=1,2,\ldots,7$.

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Example 5.1