Cubature and reconstruction of smooth isotropic random 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.