Numerical Differentiation of Functions of Two Variables Using Chebyshev Polynomials

TL;DR

This paper presents a new Chebyshev polynomial-based method for numerically differentiating bivariate functions, providing explicit error estimates and a truncation rule tailored to noise and smoothness parameters.

Contribution

It introduces a novel truncation approach using Chebyshev polynomials and hyperbolic cross for partial derivatives of bivariate functions in weighted Wiener classes.

Findings

Explicit error estimates in weighted integral norms and uniform metric.

A truncation parameter choice rule based on noise level and function smoothness.

Effective reconstruction of derivatives of arbitrary order.

Abstract

We investigate the problem of numerical differentiation of bivariate functions from weighted Wiener classes using Chebyshev polynomial expansions. We develop and analyze a new version of the truncation method based on Chebyshev polynomials and the idea of hyperbolic cross to reconstruct partial derivatives of arbitrary order. The method exploits the approximation properties of Chebyshev polynomials and their natural connection to weighted spaces through the Chebyshev weight function. We derive a choice rule for the truncation parameter as a function of the noise level, smoothness parameters of the function class, and the order of differentiation. This approach allows us to establish explicit error estimates in both weighted integral norms and uniform metric.