Estimates for nn-widths of sets of smooth functions on the torus TdTd

In this paper, we investigate nn-widths of multiplier operators Λ={λk}k∈Zd and Λ∗={λk∗}k∈Zd, Λ,Λ∗:Lp(Td)→Lq(Td)Λ,Λ∗:Lp(Td)→Lq(Td) on the dd-dimensional torus TdTd, where λk=λ(|k|) and λk∗=λ(|k|∗) for a function λλ defined on the interval [0,∞)[0,∞), with |k|=(k12+⋯+kd2)1/2 and |k|∗=max1≤j≤d|kj|. In the first part, upper and lower bounds are established for nn-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>0γ,r>0 and ξ≥0ξ≥0. We have that Λ(1)UpΛ(1)Up and Λ∗(1)Up are sets of finitely differentiable functions on TdTd, in particular, Λ(1)UpΛ(1)Up and Λ∗(1)Up are Sobolev-type classes if ξ=0ξ=0, and Λ(2)UpΛ(2)Up and Λ∗(2)Up are sets of infinitely differentiable (01r>1) functions on TdTd, where UpUp denotes the closed unit ball of Lp(Td)Lp(Td). In particular, we prove that, the estimates for the Kolmogorov nn-widths dn(Λ(1)Up,Lq(Td))dn(Λ(1)Up,Lq(Td)), dn(Λ∗(1)Up,Lq(Td)), dn(Λ(2)Up,Lq(Td))dn(Λ(2)Up,Lq(Td)) and dn(Λ∗(2)Up,Lq(Td)) are order sharp in various important situations.