Decay estimate of bivariate Chebyshev coefficients for functions with limited smoothness

TL;DR

This paper derives decay bounds for Chebyshev series coefficients of functions with limited smoothness, providing insights into approximation errors for functions of bounded variation on the unit square.

Contribution

It generalizes a key identity relating exact and approximate Chebyshev coefficients and establishes asymptotic $L^1$-approximation errors for functions with finite Vitali variation.

Findings

Decay bounds for Chebyshev coefficients of functions with finite Vitali variation.

Asymptotic $L^1$-approximation error estimates for functions of bounded variation.

Generalization of the identity relating exact and approximate Chebyshev coefficients.

Abstract

We obtain the decay bounds for Chebyshev series coefficients of functions with finite Vitali variation on the unit square. A generalization of the well known identity, which relates exact and approximated coefficients, obtained using the quadrature formula, is derived. Finally, an asymptotic $L^1$-approximation error of finite partial sum for functions of bounded variation in sense of Vitali as well as Hardy-Krause, on the unit square is deduced.