Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

  • Authors:

  • Chen Guanggui;Fang Gensun

  • Affiliations:

  • Department of Mathematics, Beijing Normal University, Beijing 100875, China;Department of Mathematics, Beijing Normal University, Beijing 100875, China

  • Venue:
  • Journal of Complexity
  • Year:
  • 2004

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Abstract

We present sharp bounds on the Kolmogorov probabilistic (N, δ)-width and p average N- width of multivariate Sobolev space with mixed derivative MW2r(Td), r = (r1 ..., rd), 1/2 r1 = ... = rv rv + 1 ≤ ... ≤ rd equipped with a Gaussian measure µ in Lq (Td), that is dN, δ (MW2r(Td)), µ, Lq (Td)) = (N-1 lnv-1 N)r1 ċ (ρ - 1)/2 (ln(v-1)/2 N) √ 1 + (1/N)ln(1/δ), dN(a) (MW2r(Td), µ, Lq (Td))p = (N-1 lnv-1 N)r1 ċ (ρ - 1)/2 (ln(v-1)/2 N), where 1 q 1 is depending only on the eigenvalues of the correlation operator of the measure µ (see (4)).