Sampling numbers of periodic Sobolev spaces with a Gaussian measure in the average case setting

Abstract

In this paper, we investigate p-average sampling numbers of a periodic Sobolev space W2r with a Gaussian measure in the Lq metric for 1≤q≤∞ and 0<p<∞, and obtain their asymptotic orders. Moreover, we show that in the average case setting, the operators In, which are the Lagrange interpolating operators, are asymptotically optimal in the Lq metric for all 1≤q≤∞. It is interesting to note that in the worst case setting, In are not asymptotically optimal algorithms in the Lq metric for q=1 or ∞.