Average case optimal algorithms in Hilbert spaces
Journal of Approximation Theory
Information-based complexity
On the average complexity of multivariate problems
Journal of Complexity
The computational complexity of differential and integral equations: an information-based approach
The computational complexity of differential and integral equations: an information-based approach
Explicit cost bounds of algorithms for multivariate tensor product problems
Journal of Complexity
Complexity and information
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers
Foundations of Computational Mathematics
Lattice rule algorithms for multivariate approximation in the average case setting
Journal of Complexity
On the power of standard information for multivariate approximation in the worst case setting
Journal of Approximation Theory
Linear information versus function evaluations for L2-approximation
Journal of Approximation Theory
On the power of standard information for multivariate approximation in the worst case setting
Journal of Approximation Theory
On the approximation of smooth functions using generalized digital nets
Journal of Complexity
Tractability results for weighted Banach spaces of smooth functions
Journal of Complexity
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We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L"~ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper. We show that the L"~ approximation problem in the worst case setting is related to the weighted L"2 approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter @a1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-^(^@a^-^1^)^/^2 and n^-^1^/^2, respectively. We also study tractability of L"~ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L"~ approximation in terms of tractability conditions of the weighted L"2 approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.