On the approximation by weighted ridge functions

TL;DR

This paper characterizes the optimal $L_{2}$ approximation of multivariate functions using weighted ridge functions, providing explicit formulas in the case of constant weights for best approximation and error estimation.

Contribution

It offers a theoretical characterization of the best weighted ridge function approximation and explicit formulas for the unweighted case.

Findings

Explicit formulas for best approximation with constant weights

Characterization of approximation error in weighted ridge functions

Theoretical insights into multivariate function approximation

Abstract

We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit formulas for both the best approximation and approximation error.