Tensor power sequences and the approximation of tensor product 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 $w_d$.