Exponential Convergence-(t,s)-Weak Tractability of Approximation in Weighted Hilbert Spaces

Abstract

We study L2-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters γj and αj for j∈N. We consider algorithms using the class of arbitrary linear functionals. We discuss the exponential convergence-(t,s)-weak tractability of these L2-approximation problems. In particular, we obtain the sufficient and necessary conditions on the weights for exponential convergence-weak tractability and exponential convergence-(t,1)-weak tractability with t<1.