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Estimates for [formula omitted]-widths of sets of smooth functions on the torus [formula omitted]

Authors
Journal
Journal of Approximation Theory
0021-9045
Publisher
Elsevier
Volume
183
Identifiers
DOI: 10.1016/j.jat.2014.03.014
Keywords
  • Torus
  • Width
  • Multiplier
  • Smooth Function

Abstract

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|∗=max1≤j≤d|kj|. 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)Up and Λ∗(1)Up are sets of finitely differentiable functions on Td, in particular, Λ(1)Up and Λ∗(1)Up are Sobolev-type classes if ξ=0, and Λ(2)Up and Λ∗(2)Up are sets of infinitely differentiable (0<r<1) or analytic (r=1) or entire (r>1) functions on Td, where Up 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.

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