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Approximating periodic functions in Hölder type metrics by Fourier sums and Riesz means

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Abstract

Let M be a fixed space of 2π-periodic functions, Lp or C, let ωr(f, h) be the continuity modulus of order r of the function f in the space M, and let ϕ(t) be a function such that ϕ(t) > 0 for t > 0. By Sn(f) we denote the Fourier sums and by Rn,r(f) we denote the Riesz sums (the Fejér sums for r = 1) of the function f. Set

$$K_m (f) = K_{m,\varphi } (f) = \mathop {\sup }\limits_{0 < v < \infty } \frac{{\omega _m (f,v)}}{{\varphi (v)}}$$

. The paper studies the dependence of the behavior of the quantities

$$K_m (f - S_n (f)) and K_m (f - R_{n,r} (f))$$

as n → ∞ on the structural properties of the function f expressed in terms of the continuity moduli. In this way, general results are established, which are applicable to other approximation methods as well. Bibliography: 10 titles.

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References

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Authors and Affiliations

  1. St.Petersburg State University, St.Petersburg, Russia

    V. V. Zhuk

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  1. V. V. Zhuk
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Correspondence to V. V. Zhuk.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 70–88.

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Zhuk, V.V. Approximating periodic functions in Hölder type metrics by Fourier sums and Riesz means. J Math Sci 150, 2045–2055 (2008). https://doi.org/10.1007/s10958-008-0121-1

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  • DOI: https://doi.org/10.1007/s10958-008-0121-1

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