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 L2 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 ε-approximation in the d-variate case grows faster than any polynomial in ε−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 ε−t and ds. We restrict our attention to product weights, defined as products of powers of weightlets γj. 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≤1, then our problem is (s,t)-weakly tractable iff γj=o(j−(2−rs)∕(rs)) as j→∞.