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

Authors

  • A.S. Serdyuk Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine
  • I.V. Sokolenko Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine https://orcid.org/0000-0002-8534-4616
https://doi.org/10.15330/cmp.15.1.246-259

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 φ21, with fixed generated kernels Ψˉβ, which have Fourier series of the form k=1ψ(k)cos(ktβkπ/2), where ψ(k)0, ψ2(k)<,βkR. It is shown that for rapidly decreasing sequences ψ(k) (in particular, if limkψ(k+1)/ψ(k)=0) 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.