Linear information versus function evaluations for L2-approximation

• Authors: Aicke Hinrichs;Erich Novak;Jan Vybíral
• Venue: Journal of Approximation Theory
• Year: 2008

We study algorithms for the approximation of functions, the error is measured in an L"2 norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n^-^b. Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether pb is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Wozniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that p12 implies b=2p^2/(2p+1)p-12. Here we prove that the case p=12 and b=0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case p12 is quite different, it is still open whether b=p always holds in that case.