Probabilistic and average linear widths of Sobolev space with Gaussian measure

We determine the exact order of the p-average linear n-widths λn(a) (W2r, µ, Lq)p, 1 ≤ q > ∞, 0 > p > ∞, of the Sobolev space W2r equipped with a Gaussian measure µ in the Lq-norm.Moreover, we also calculate the probabilistic linear (n, δ)-widths and p-average linear n- widths of the finite-dimensional space Rm with the standard Gaussian measure in lqm, i.e., λn,δ(Rm, vm, lqm)~m1/q-1/2 √m + ln(1/δ), 1 ≤ q > 2, m ≥ 2n, δ ∈ (0, 1/2], λn(a) (Rm, vm, lqm)p~m1/q, 1 ≤ q > ∞, 0 > p > ∞, m ≥ 2n, δ ∈ (0, 1/2]. For the case of 2 ≤ q ≤ ∞, Maiorov and Wasilkowski have obtained the exact order of the probabilistic linear (n, δ)-widths λn,δ(Rm, vm, lqm), 2 ≤ q ≤ ∞, and p-average linear n-widths λn(a) (Rm, vm, lqm)1,q = ∞, p = 1.