Tensor power sequences and the approximation of tensor product operators

TL;DR

This paper investigates the asymptotic behavior of tensor power sequences and their role in approximating tensor product operators, extending known results to more general polynomial decay sequences.

Contribution

It provides new asymptotic and preasymptotic results for tensor powers of arbitrary polynomial decay sequences, applicable to various tensor product operator embeddings.

Findings

Asymptotic behavior characterized for tensor powers of polynomial decay sequences

Results applicable to Sobolev and Jacobi function embeddings

Extension of approximation number analysis to broader tensor operators

Abstract

The approximation numbers of the $L_2$-embedding of mixed order Sobolev functions on the $d$-torus are well studied. They are given as the nonincreasing rearrangement of the $d$-th tensor power of the approximation number sequence in the univariate case. I present results on the asymptotic and preasymptotic behavior for tensor powers of arbitrary sequences of polynomial decay. This can be used to study the approximation numbers of many other tensor product operators, like the embedding of mixed order Sobolev functions on the $d$-cube into $L_2\left([0,1]^d\right)$ or the embedding of mixed order Jacobi functions on the $d$-cube into $L_2\left([0,1]^d,w_d\right)$ with Jacobi weight $w_d$.