Numerical Differentiation of Functions of Two Variables Using Chebyshev Polynomials

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.

💡 Research Summary

The paper addresses the ill‑posed problem of numerically differentiating bivariate functions that belong to weighted Wiener classes. The authors develop a regularized differentiation scheme based on truncated Chebyshev polynomial expansions combined with a hyperbolic‑cross index set.

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