A note on the equioscillation theorem for best ridge function approximation

TL;DR

This paper extends the classical equioscillation theorem to the context of best approximation of continuous functions by sums of two ridge functions in multiple dimensions, providing a necessary and sufficient condition.

Contribution

It introduces a new equioscillation theorem for ridge function approximation, generalizing classical polynomial results to multivariate settings.

Findings

Derived a necessary and sufficient condition for best ridge function approximation.

Connected ridge approximation to classical Chebyshev equioscillation theorem.

Enhanced understanding of multivariate function approximation methods.

Abstract

We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.