Linear widths of a multivariate function space equipped with a Gaussian measure

  • Authors:

  • Chen Guanggui;Fang Gensun

  • Affiliations:

  • School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China and School of Computers and Mathematical-Physical Science, XiHua University, Chengdu, China;School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2005

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Abstract

We determine the asymptotic values on the linear probabilistic (N,δ)-widths and linear p-average N-widths of the space of multivariate functions with bounded mixed derivative MW2r(Td), r = (r1 ..... rd), 1/2 r1=...=rvrv+1≤ ... ≤rd equipped with a Gaussian measure µ in Lq (Td). That is, the following asymptotic equivalences hold: (1) If 1q≤2, then λN,δ(MW2r(Td),µ, Lq(Td)) = (N-1 lnv-1N)r1+(ρ-1)/2(ln(v-1)/2N) × √1+(1/N)ln(1/δ). (2) If 1 q N(a)(MW2r(Td),µ, Lq(Td)) = (N-1 lnv-1N)r1+(ρ-1/2)(ln(v-1)/2N). Here 0 1 depends only on the eigenvalues of the correlation operator of the measure µ (see (4)).If the dimension d ≥ 2, then the asymptotic exact order of probabilistic linear widths of MWr2(Td with the Gaussian measure µ in the Lq(Td) space for the cases q = 1, 2 q ≤ ∞ and the average linear widths λN(a) (MWr2(Td), µ, Lq(Td)) for the case q = 1 and q = ∞ are still open.