Nonlinear tensor product approximation of functions

TL;DR

This paper investigates nonlinear tensor product approximation of multivariate functions, extending classical bilinear approximation to higher dimensions with new results on best approximation under mixed smoothness.

Contribution

It provides new theoretical results on best multilinear approximation in $L_p$ spaces for functions with mixed smoothness, advancing understanding beyond bilinear cases.

Findings

Results on decay rates of approximation errors

Extension of bilinear approximation to higher dimensions

Analysis under mixed smoothness assumptions

Abstract

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel $f(x_1,x_2)$. There are known results on the rate of decay of errors of best bilinear approximation in $L_p$ under different smoothness assumptions on $f$. The problem of multilinear approximation (nonlinear tensor product approximation) in the case $d\ge 3$ is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in $L_p$ under mixed smoothness assumption on $f$.