Information of varying cardinality
Journal of Complexity
Information-based complexity
Journal of Approximation Theory
An integrated fractional Fourier transform
Journal of Computational and Applied Mathematics
Asymptotically optimal weighted numerical integration
Journal of Complexity
Optimal integration of Lipschitz functions with a Gaussian weight
Journal of Complexity
Complexity of weighted approximation over R
Journal of Approximation Theory
Delayed curse of dimension for Gaussian integration
Journal of Complexity
New averaging technique for approximating weighted integrals
Journal of Complexity
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We study weighted approximation and integration of Gaussian stochastic processes X defined over R+ whose rth derivatives satisfy a Hölder condition with exponent β in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding ε. We provide sufficient conditions in terms of the weight and the parameters r and β for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0, 1 ], i.e., proportional to ε-1/α where α = r + β for the approximation and α = r + β + 1/2 for the integration.