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 L p , 1 ≤ p < ∞, in the case where the sequence ψ(k) is such that the product ψ(n)n 1/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 ‖f ψ β ‖1 ≤ 1.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 7, pp. 916–936, July, 2015.
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Serdyuk, A.S., Stepanyuk, T.A. Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness. Ukr Math J 67, 1038–1061 (2015). https://doi.org/10.1007/s11253-015-1134-9
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DOI: https://doi.org/10.1007/s11253-015-1134-9