Chebyshev approximation for multivariate functions

TL;DR

This paper extends Chebyshev approximation theory to multivariate functions using convexity and nonsmooth analysis, providing optimality conditions where traditional alternance concepts are not straightforward to apply.

Contribution

It introduces a novel approach for multivariate Chebyshev approximation based on convexity and nonsmooth analysis, overcoming challenges in extending univariate alternance theory.

Findings

Derived optimality conditions for multivariate Chebyshev approximation

Proposed an alternative approach using convexity and nonsmooth analysis

Addressed the extension of alternance concept to multivariate functions

Abstract

In this paper, we derive optimality conditions (Chebyshev approximation) for multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions is very elegant. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not very straightforward, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.