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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received: 21.09.1984

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
    Related articles on Google Scholar: Russian articles, English articles

    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  mathscinet  zmath  isi
    2. Rusak V., “Precise Orders of Best Rational-Approximations for Convolutions of Weyl Kernels and Functions From Lp”, 315, no. 2, 1990, 313–316  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    6. Mitenkov V., Rusak V., “Error Estimates for an Approximation to a Characteristic Singular Integral Equation”, Differ. Equ., 37:3 (2001), 439–443  mathnet  crossref  mathscinet  zmath  isi
    7. A. P. Starovoitov, “Rational Approximations of Riemann–Liouville and Weyl Fractional Integrals”, Math. Notes, 78:3 (2005), 391–402  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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