Estimates of deviations of Fourier sums on Weyl-Nagy classes $$$W^r_{\beta,1}$$$

A.S. Serdyuk; I.V. Sokolenko

doi:10.15421/242521

Estimates of deviations of Fourier sums on Weyl-Nagy classes $$$W^r_{\beta,1}$$$

A.S. Serdyuk (Institute of Mathematics of NAS of Ukraine), https://orcid.org/0000-0003-2659-8920
I.V. Sokolenko (Institute of Mathematics of NAS of Ukraine), https://orcid.org/0000-0002-8534-4616

Abstract

We establish estimates for exact upper bounds of deviations of partial Fourier sums $$$S_{n-1}(f)$$$ on classes $$$W^r_{\beta,1}, r>2, \beta\in\mathbb{R},$$$ of $$$2\pi$$$-periodic functions whose $$$(r,\beta)$$$-derivatives in the Weyl-Nagy sense belong to the unit ball of the space $$$L_1$$$. The specified estimates allow us to write asymptotic equalities for the quantities $$$\sup\limits_{f\in W^r_{\beta,1}}|f(x)-S_{n-1}(f;x)|$$$ as $$$n\rightarrow\infty$$$, $$$r\rightarrow\infty$$$ for arbitrary relations between the parameters $$$r$$$ and $$$n$$$.

Keywords

Fourier sum; Weil-Nagy class; asymptotic equality

MSC 2020

Pri 42A10

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References

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