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In this paper, we study average sampling numbers of the multivariate periodic function space L ̊ 2 with a Gaussian measure μ in the L q metric for 1 ≤ q ≤ ∞ , and obtain their asymptotical orders, where the Cameron–Martin space of the measure μ is an anisotropic periodic Sobolev space. Moreover, we show that in the average case setting, the Lagrange interpolating operators are asymptotically optimal linear algorithms in the L q metric for all 1 ≤ q ≤ ∞ . This is different from the situation in the worst case setting, where the Lagrange interpolating operators are not asymptotically optimal linear algorithms in the L q metric for q = 1 or ∞ . [ABSTRACT FROM AUTHOR]

We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions. [ABSTRACT FROM AUTHOR]

We study approximation of multivariate periodic functions from Besov and Triebel–Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the L q -norm for functions from the Besov spaces B p , θ s (T d) and the Triebel–Lizorkin spaces F p , θ s (T d) for all s > 0 and admissible 1 ≤ p , θ ≤ ∞ as well as provide analogues of the Littlewood–Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators. [ABSTRACT FROM AUTHOR]