Integration and approximation in arbitrary dimensions

We study multivariate integration in the worst case setting and multivariate approximation in the average case setting for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in checking when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of " is bounded by C(d)" ?p for some exponent p and some function C(d). Strong tractability means that C(d) can be made independent of d. The "-exponents of tractability and strong tractability are deened as the smallest powers of " ?1 in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the "-exponents, and for some cases we nd the exact values of these "-exponents. For some weighted Korobov and Sobolev spaces, the strong "-exponent is the same as the exponent for d = 1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals. Our main result is that the "-exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L 1 norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non shift-invariant kernel, we construct the corresponding space with shift-invariant kernel and show that 1 integration and approximation for the non shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces.