Integration and L₂-approximation: Average case setting with isotropic wiener measure for smooth functions

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 Θ(n^{−(d+4r+1)/(2d)}) and Θ(n^{−(4r+1)/(2d)}) for the integration and L₂-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 L₂-approximation problem is intractable. The integration and L₂-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.