Integration and $L_2$-Approximation: Average Case Setting with Isotropic Wiener Measure for Smooth Functions
Klaus Ritter, G.W. Wasilkowski
. We propose isotropic probability measures defined on classes of smooth multivariate functions. These provide a natural extension of the classical isotropic Wiener measure to multivariate functions from C 2r . We show that in the corresponding average case setting the minimal errors of algorithms that use n function values are \\Theta \\Gamma n \\Gamma(d+4r+1)=2d \\Delta and \\Theta \\Gamma n \\Gamma(4r+1)=2d \\Delta for the integration and L 2 -approximation problems respectively. Here d is the number of variables of the corresponding class of functions. This means that the minimal average errors depend essentially on the number d of variables. In particular, for d large relative to r, the L 2 -approximation problem is intractable. The integration and L 2 -approximation problems have been recently studied with measures whose covariance kernels are tensor products. The results for these measures and for isotropic measures differ significantly. 1 Introduction We study the integratio...
Rocky Mountain Journal of Mathematics•Dec 1, 1996•10 citations