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

We study L_2-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 ⱼ and ⱼ for j=1,\,2,,\,d. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the exponential convergence-(t,s)-weak tractability (EC-(t,s)-WT) of these L_2-approximation problems under the absolute or normalized error criterion. In particular, we obtain the sufficient and necessary conditions for EC-(1,1)-WT and EC-(t,1)-WT with t