Tractability of multivariate approximation over weighted standard Sobolev spaces

Abstract

We study multivariate approximation for complex-valued functions defined over the $d$-dimensional torus, these functions belonging to weighted standard Sobolev spaces of smoothness $r$. Algorithms are allowed to use finitely many arbitrary continuous linear functionals, which may be chosen adaptively. The error of an algorithm is measured by the $L_2$ norm, in the worst case setting. No matter how we choose positive weights, we prove that multivariate approximation cannot be quasi-polynomially tractable, meaning that the minimal number of continuous linear functionals needed to find an $\epsilon$-approximation in the $d$-variate case grows faster than any polynomial in ${\epsilon }^{-lnd}$. We also study $(s,t)$-weak tractability for positive $s$ and  $t$, meaning that the minimal number of continuous linear functionals is not exponential in  ${\epsilon }^{-t}$ and $d^s$. We restrict our attention to product weights, defined as products of powers of weightlets ${\gamma }_j$. We obtain conditions on the weightlets that are necessary and sufficient for our problem to be $(s,t)$-weakly tractable. In particular, if $rs<2$ and $t\le 1$, then our problem is $(s,t)$-weakly tractable iff ${\gamma }_j=o\left(j^{-(2-rs)∕\left(rs\right)}\right)$ as $j\to \infty$.