Alexander Kushpel

Estimates of n-widths of Sobolev's classes on compact globally symmetric spaces of rank one

Estimates of Kolmogorovs and linear n-widths of Sobolevs classes on compact globally symmetric spaces of rank 1 (i.e. on Sd, Pd(R), Pd(C), Pd(H), P16(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. New estimates for the (p,q)-norms of multiplier operators Λ={λk}k∈N are given. We apply our results to get sharp orders of best polynomial approximation and n-widths.

On the norm of the Fourier-Jacobi projection

We extend the results of Pollard [7] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted Lp space.

A multiplier version of the Bernstein inequality on the complex sphere

We prove a multiplier version of the Bernstein inequality on the complex sphere. Included in this is a new result relating a bivariate sum involving Jacobi polynomials and Gegenbauer polynomials, which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group, with the reproducing kernel of the sum of these spaces, which is irreducibly invariant under the action of the unitary group.

Estimates for n-widths of sets of smooth functions on the torus Td

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 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.