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

In this paper, we study average sampling numbers of the multivariate periodic function space L̊2 with a Gaussian measure μμ in the LqLq metric for 1≤q≤∞1≤q≤∞, and obtain their asymptotical orders, where the Cameron–Martin space of the measure μμ 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 LqLq metric for all 1≤q≤∞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 LqLq metric for q=1q=1 or ∞∞.