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Type: Artigo de periódico
Title: Estimates For N-widths Of Sets Of Smooth Functions On The Torus Td
Author: Kushpel A.
Stabile R.L.B.
Tozoni S.A.
Abstract: In this paper, we investigate n-widths of multiplier operators Λ={λk}k∈Zd and Λ*={λk*}k∈Zd, Λ,Λ*:Lp(Td)→Lq(Td) on the d-dimensional torus Td, where λk = λ (|k|) and λk*=λ(|k|*) for a function λ defined on the interval [0, ∞), with |k|=(k12+⋯+kd2)1/2 and |k| * = max 1 ≤j ≤d|k j|. In the first part, upper and lower bounds are established for n-widths of general multiplier operators. In the second part, we apply these results to the specific multiplier operators Λ(1)={|k|-γ(ln|k|)-ξ}k∈Zd, Λ*(1)={|k|*-γ(ln|k|*)-ξ}k∈Zd, Λ(2)={e-γ|k|r}k∈Zd and Λ*(2)={e-γ|k|*r}k∈Zd for γ, r > 0 and ξ ≥ 0. We have that Λ (1) U p and Λ*(1)Up are sets of finitely differentiable functions on Td, in particular, Λ (1) U p and Λ*(1)Up are Sobolev-type classes if ξ = 0, and Λ (2) U p and Λ*(2)Up are sets of infinitely differentiable (0 < r < 1) or analytic (r = 1) or entire (r > 1) functions on Td, where U p denotes the closed unit ball of Lp(Td). In particular, we prove that, the estimates for the Kolmogorov n-widths dn(Λ(1)Up,Lq(Td)), dn(Λ*(1)Up,Lq(Td)), dn(Λ(2)Up,Lq(Td)) and dn(Λ*(2)Up,Lq(Td)) are order sharp in various important situations. © 2014 Elsevier Inc.
Editor: Academic Press Inc.
Rights: fechado
Identifier DOI: 10.1016/j.jat.2014.03.014
Date Issue: 2014
Appears in Collections:Unicamp - Artigos e Outros Documentos

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