Ukrainian Mathematical Journal

, Volume 67, Issue 7, pp 1038–1061 | Cite as

Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

  • A. S. Serdyuk
  • T. A. Stepanyuk
Article

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of 2π-periodic functions whose (ψ,\( \beta \))-derivatives belong to unit balls in the spaces Lp, 1 ≤ p < ∞, in the case where the sequence ψ(k) is such that the product ψ(n)n1/p may tend to zero slower than any power function and \( {\displaystyle {\sum}_{k=1}^{\infty }}{\psi}^{p\prime }{(k)}^{p\prime -2}<\infty \kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 1<p<\infty, \frac{1}{p}+\frac{1}{p^{\prime }}=1,\kern0.5em \mathrm{o}\mathrm{r}{\displaystyle {\sum}_{k=1}^{\infty }}\psi (k)<\infty \) for p = 1. Similar estimates are also established in the Ls-metrics, 1 < s ≤ ∞, for the classes of summable (ψ,\( \beta \))-differentiable functions such that ‖fβψ1 ≤ 1.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. S. Serdyuk
    • 1
  • T. A. Stepanyuk
    • 2
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.L. Ukrainka East-European National UniversityLutskUkraine

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