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Numerical Differentiation of Functions of Two Variables Using Chebyshev Polynomials

Maksym Kyselov, Sergiy G. Solodky

TL;DR

This work addresses the ill-posed problem of numerically differentiating bivariate functions from weighted Wiener classes by employing a truncated Fourier-Chebyshev expansion on $Q=[-1,1]^2$ with a hyperbolic-cross domain. The authors derive explicit error bounds for recovering partial derivatives of arbitrary order in multiple norms ($L_{2,\omega}$, $C$, and $L_{q,\omega}$) and establish regularization via a truncation parameter $n$ tied to the noise level $\delta$ and smoothness parameters, yielding concrete rates such as $O\big(\delta^{(\mu_1-2r+1/s-1/2)/(\mu_1-1/p+1/s)}\big)$ in $L_{2,\omega}$ and analogous rates in other metrics. The results are obtained for function classes $BW^{\overline{\mu}}_{s,2}$ and demonstrate how the hyperbolic cross truncation and Chebyshev basis achieve sharp (order) error estimates, with advantages in the $C$-metric over Legendre-based methods. The work also discusses optimality, the cardinality of the truncation domain, and practical parameter choices, providing a comprehensive framework for stable, high-order bivariate differentiation from noisy data. This advances regularized numerical differentiation in weighted spaces and informs algorithmic choices for reliable derivative recovery in applications requiring two-variable differentiation.

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.

Numerical Differentiation of Functions of Two Variables Using Chebyshev Polynomials

TL;DR

This work addresses the ill-posed problem of numerically differentiating bivariate functions from weighted Wiener classes by employing a truncated Fourier-Chebyshev expansion on with a hyperbolic-cross domain. The authors derive explicit error bounds for recovering partial derivatives of arbitrary order in multiple norms (, , and ) and establish regularization via a truncation parameter tied to the noise level and smoothness parameters, yielding concrete rates such as in and analogous rates in other metrics. The results are obtained for function classes and demonstrate how the hyperbolic cross truncation and Chebyshev basis achieve sharp (order) error estimates, with advantages in the -metric over Legendre-based methods. The work also discusses optimality, the cardinality of the truncation domain, and practical parameter choices, providing a comprehensive framework for stable, high-order bivariate differentiation from noisy data. This advances regularized numerical differentiation in weighted spaces and informs algorithmic choices for reliable derivative recovery in applications requiring two-variable differentiation.

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.
Paper Structure (4 sections, 14 theorems, 108 equations)

This paper contains 4 sections, 14 theorems, 108 equations.

Table of Contents

  1. Introduction. Description of the problem
  2. Truncation method. Error estimation in $L_{2,\omega}$ metric
  3. Error estimate for truncation method in the metric of $C$
  4. Truncation method. Error estimation in $L_{q,\omega}$ metric

Key Result

Lemma 2.1

Let $f\in W^{\overline{\mu}}_{s,2}$, $1\leq s< \infty$, $\mu_1>2r-1/s+1/2$, $\mu_2>\mu_1-2r$. Then for $1\leq \gamma < \frac{\mu_2+1/s-1/2}{\mu_1-2r+1/s-1/2}$ it holds

Theorems & Definitions (17)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 4.1
  • Lemma 4.2
  • ...and 7 more