Decay analysis of bivariate Chebyshev coefficients for functions with limited regularity

Abstract

The Chebyshev polynomial approximation is a useful tool to approximate smooth and non-smooth functions. In fact, for a sufficiently smooth function, the partial sum of Chebyshev series expansion provides optimal polynomial approximation. Moreover, because the construction of these polynomial approximations is computational efficient, they are widely used in numerical schemes for solving partial deferential equations. Significant efforts have been devoted to establishing decay bounds for series coefficients, including Chebyshev, Jacobi, and Legendre series, for both smooth and non-smooth univariate functions. However, the literature lacks similar estimates for bivariate functions. This paper aims to address this gap by examining the decay estimates of bivariate Chebyshev coefficients, contributing both theoretically and practically to the understanding and application of Chebyshev series expansions, especially concerning functions with limited smoothness. Additionally, we derive $L^1$-error estimates for the partial sum of Chebyshev series expansions of functions with bounded Vitali variation. Furthermore, we provide an estimate for the discrepancy between exact and approximated Chebyshev coefficients, leveraging a quadrature formula. This analysis leads to the deduction of an asymptotic $L^1$-approximation error for finite partial sums of Chebyshev series with approximated coefficients.