Asymptotic estimates for the widths of classes of functions of high smothness

Keywords:
Bernstein width, Kolmogorov width, linear width, projection width, Fourier sum, Weyl-Nagy class, class of the generalized Poisson integrals, (ψ,ˉβ)-integral, asymptotic equality
Published online:
2023-06-30
Abstract
We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of 2π-periodic functions φ, such that ‖φ‖2≤1, with fixed generated kernels Ψˉβ, which have Fourier series of the form ∞∑k=1ψ(k)cos(kt−βkπ/2), where ψ(k)≥0, ∑ψ2(k)<∞,βk∈R. It is shown that for rapidly decreasing sequences ψ(k) (in particular, if lim) the obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.
How to Cite
(1)
Serdyuk, A.; Sokolenko, I. Asymptotic Estimates for the Widths of Classes of Functions of High Smothness. Carpathian Math. Publ. 2023, 15, 246-259.