Averaged characteristics of smoothness in L2 and estimations for the widths values of function classes

Authors

  • S. Vakarchuk Alfred Nobel University, Dnipro
  • V. Zabutna Oles Honchar Dnipro National University
  • M. Vakarchuk Oles Honchar Dnipro National University

DOI:

https://doi.org/10.3842/umzh.v77i2.8794

Keywords:

generalized finite difference operator, generalized smoothness characteristic of function, Fourier series, Fourier coefficients, majorant, widths of function classes.

Abstract

UDC 517.5

In the space of  2π-periodic functions L2, we investigate the characteristic of smoothness  ω∗M(f,t):= (1/t)∫t0ωM(f,τ)dτ obtained as a result of averaging of the generalized modulus of continuity ωM(f)  formed by using a generalized finite-difference operator ΔMh:L2→L2. We also study some properties of the functions ωM(f) and ω∗M(f). For the classes of functions W(ω∗M,Φ), where Φ is a majorant, we determine the lower and upper estimates for the values of a series of  n-widths  and the establish the condition for Φ under which we obtain the exact values of these estimates. Several exact results are illustrated by specific examples.

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28.03.2025

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Vol. 77 No. 2 (2025)

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Copyright (c) 2025 Сергій Вакарчук, Валентина Забутна, Михайло Вакарчук

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How to Cite

Vakarchuk, S., et al. “Averaged Characteristics of Smoothness in L2 and Estimations for the Widths Values of Function Classes”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 2, Mar. 2025, pp. 83–106, https://doi.org/10.3842/umzh.v77i2.8794.