Average sampling numbers of multivariate periodic function spaces with a Gaussian measure

Abstract

In this paper, we study average sampling numbers of the multivariate periodic function space L̊2 with a Gaussian measure μ in the Lq 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 Lq 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 Lq metric for q=1 or ∞.