Probabilistic linear （N, δ）-widths and p-averagelinear N-widths of Sobolev space W2^r（T）, equipped with a Gaussian probability measure #, are studied in the metric of Sq （T） （1 ≤ Q ≤∞）, and determined the...
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Probabilistic linear （N, δ）-widths and p-averagelinear N-widths of Sobolev space W2^r（T）, equipped with a Gaussian probability measure #, are studied in the metric of Sq （T） （1 ≤ Q ≤∞）, and determined the asymptotic equalities：
λN,δ（W2^r（T）,μ,Sq（T））={（N^-1）^r＋p/2-1/q√1＋1/N·ln1/δ, 1≤q≤2,
（N^-1）^r＋p/2-1/q（1＋N^-1/q√ln1/δ）,2〈q〈∞,
（N^-1）^r＋p/2√lnN/δ, q=∞,
and
λN^（a）（W2^r（T）,μ,Sq（T））p={（N^-1）^r＋p/2-1/q, 1≤q〈∞,
（N^-1）^r＋p/2-1/q√lnN, q=∞,
where 0 〈 p 〈 ∞, δ∈ （0, 1/2], ρ 〉 1, and Sq（T） is a subspace of L1（T）, in which the Fourier series is absolutely convergent in lq sense.
We determine the asymptotic values on the linear probabilistic (N, delta)-widths and linear p-average N-widths of the space of multivariate functions with bounded mixed derivative MW2r(T-d), r = (r(1),..., r(d)), 1/2 ...
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We determine the asymptotic values on the linear probabilistic (N, delta)-widths and linear p-average N-widths of the space of multivariate functions with bounded mixed derivative MW2r(T-d), r = (r(1),..., r(d)), 1/2 < r(1) = ... = r(v) < r(v+1) less than or equal to ... less than or equal to r(d), equipped with a Gaussian measure mu in L-q(T-d). That is, the following asymptotic equivalences hold: (1) If 1 < q less than or equal to 2, then lambda(N,delta) (MW2r (T-d), mu, L-q(T-d)) asymptotic to (N-1 ln(v-1) N)(r1+(rho-1/2)) (ln((v-1)/2) N) X root1 + (1/N) ln(1/delta). (2) If 1 < q < infinity, then lambda(N)((a)) (MW2r(T-d),mu, L-q(T-d)) asymptotic to (N-1 ln(v-1) N)(r1+(rho-1/2)) (ln((v-1)/2) N). Here 0 < delta less than or equal to 1/2, and rho > 1 depends only on the eigenvalues of the correlation operator of the measure mu (see (4)). If the dimension d greater than or equal to 2, then the asymptotic exact order of probabilistic linearwidths of MW2r(T-d) with the Gaussian measure mu in the L-q(T-d) space for the cases q = 1, 2 < q less than or equal to infinity;and the average linear widths lambda(N)((a)) (MW2r(T-d)), mu, L-q(T-d) for the cases q = 1 and q = infinity are still open. (C) 2004 Elsevier Inc. All rights reserved.
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