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 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 4(12), Pages 492–515 (Mi msb2172)

Sharp order estimates for best rational approximations in classes of functions representable as convolutions

V. N. Rusak

Abstract: Let $h(t)$ be a function of bounded variation, $[\operatorname{Var}h(t)]_0^{2\pi}\leqslant1$, and $D_r(t)$ the Weyl kernel of order $r$, i.e. $D_r(t)=\sum_{k=1}^\infty k^{-r}\cos(kt-\frac {r\pi}{2})$, $r>0$. Denote by $W_{2\pi}^r V$ and $W_{2\pi}^r V_0$ the classes of functions represented by the corresponding formulas
$$f(k)=\frac{a_0}2+\frac1\pi\int_0^{2\pi}D_r(x-t)h(t) dt, \qquad f(x)=\frac1\pi\int_0^{2\pi}D_{r+1}(x-t) dh(t).$$
The conjugate classes of functions $\widetilde{W_{2\pi}^r V}$ and $\widetilde{W_{2\pi}^r V_0}$ are also considered; they are convolutions of conjugate Weyl kernels with functions of bounded variation.
The following main result is proved:
$$\sup_{f\in K^r}\mathbf R_n^T(f)\asymp\frac1{n^{r+1}},$$
where $\mathbf R_n^T(f)$ is the best uniform approximation by trigonometric rational functions of order at most $n$, and $K^r$ is one of the classes
$$W_{2\pi}^r V,\qquad W_{2\pi}^r V_0,\qquad\widetilde{W_{2\pi}^r V},\qquad\widetilde{W_{2\pi}^r V_0}.$$

Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 491–513

Bibliographic databases:

UDC: 517.51+517.53
MSC: 41A20, 42A10, 41A25

Citation: V. N. Rusak, “Sharp order estimates for best rational approximations in classes of functions representable as convolutions”, Mat. Sb. (N.S.), 128(170):4(12) (1985), 492–515; Math. USSR-Sb., 56:2 (1987), 491–513

Citation in format AMSBIB
\Bibitem{Rus85} \by V.~N.~Rusak \paper Sharp order estimates for best rational approximations in classes of functions representable as convolutions \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 4(12) \pages 492--515 \mathnet{http://mi.mathnet.ru/msb2172} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=820399} \zmath{https://zbmath.org/?q=an:0632.41010} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 2 \pages 491--513 \crossref{https://doi.org/10.1070/SM1987v056n02ABEH003048} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Rusak V., “The Best Rational-Approximations of the Weil Core Convolution and the Functions From Lp”, Dokl. Akad. Nauk Belarusi, 34:8 (1990), 681–683
2. Rusak V., “Precise Orders of Best Rational-Approximations for Convolutions of Weyl Kernels and Functions From Lp”, 315, no. 2, 1990, 313–316
3. Rusak V., Braiess D., “The Best Polynomial and Rational-Approximations of the Classes of Functions in the Integral Metric”, Dokl. Akad. Nauk Belarusi, 36:3-4 (1992), 205–208
4. Starovoitov A., “The Accurate Orders of Rational-Approximations of Reman-Lewellyas Nucleus Convolution and Functions From l(P)”, Dokl. Akad. Nauk Belarusi, 38:1 (1994), 27–30
5. Rovba E., “On the Approximation of Functions of a Limited Variation by the Freyer and Jackson Rational Operators”, Dokl. Akad. Nauk Belarusi, 42:4 (1998), 13–17
6. Mitenkov V., Rusak V., “Error Estimates for an Approximation to a Characteristic Singular Integral Equation”, Differ. Equ., 37:3 (2001), 439–443
7. A. P. Starovoitov, “Rational Approximations of Riemann–Liouville and Weyl Fractional Integrals”, Math. Notes, 78:3 (2005), 391–402
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