Tensor power sequences and the approximation of tensor product operators
Provides theoretical tools for analyzing approximation numbers of tensor product operators, which is relevant for numerical analysis and approximation theory.
The paper studies the asymptotic and preasymptotic behavior of tensor powers of sequences with polynomial decay, enabling analysis of approximation numbers for various tensor product operators beyond the well-known Sobolev embedding on the torus.
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 $w_d$.