Worst case tractability of L_2-approximation for weighted Korobov spaces
share
We study L_2-approximation problems APP_d in the worst case setting in the weighted Korobov spaces H_d,, with parameter sequences ={_j} and ={_j} of positive real numbers 1≥_1≥_2≥⋯≥ 0 and 1/2<_1≤_2≤⋯. We consider the minimal worst case error e(n,APP_d) of algorithms that use n arbitrary continuous linear functionals with d variables. We study polynomial convergence of the minimal worst case error, which means that e(n,APP_d) converges to zero polynomially fast with increasing n. We recall the notions of polynomial, strongly polynomial, weak and (t_1,t_2)-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in d and ^-1 with the accuracy of the approximation. We obtain that the matching necessary and sufficient condition on the sequences and for strongly polynomial tractability or polynomial tractability is :=lim inf_j→∞ln_j^-1/ln j>0, and the exponent of strongly polynomial tractability is p^str=2max{1/, 1/2_1}.
READ FULL TEXT