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Trudy Inst. Mat. i Mekh. UrO RAN, 2013, Volume 19, Number 2, Pages 71–78 (Mi timm933)  

This article is cited in 3 scientific papers (total in 3 papers)

Approximation of harmonic functions by algebraic polynomials on a circle of radius smaller than one with constraints on the unit circle

N. A. Baraboshkina

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: A compact expression is found for the value of the best integral approximation of the linear combination $\lambda P_r+\mu Q_r$, where $P_r$ is the Poisson kernel and $Q_r$ is its conjugate, by trigonometric polynomials of a given order in the form of a combination of the functions $\arctan$ and $\ln$. For $\mu=0$, the expression is Krein's result, and, for $\lambda=0$, it is Nagy's result. If $\lambda\mu\not=0$, the expression is much simpler than the representation in the form of a series found by Bushanskii. It is shown that, if the function of limit values on the unit circle $\Gamma$ of the real part $u=\mathrm{Re}F$ of a certain function $F=u+iv$ that is analytic inside the unit circle and such that $\|u\|_{L(\Gamma)}\le1$ is known, then the problem of the best integral approximation of the linear combination $\lambda u+\mu v$ on a concentric circle of radius $r<1$ by algebraic polynomials is reduced to the integral approximation of the kernel $\lambda P_r+\mu Q_r$ on the period $[0,2\pi)$ by trigonometric polynomials.

Keywords: best approximation, trigonometric polynomial, harmonic function, algebraic polynomial, class of convolutions, Poisson kernel.

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Bibliographic databases:
UDC: 517.51
Received: 28.01.2013

Citation: N. A. Baraboshkina, “Approximation of harmonic functions by algebraic polynomials on a circle of radius smaller than one with constraints on the unit circle”, Trudy Inst. Mat. i Mekh. UrO RAN, 19, no. 2, 2013, 71–78

Citation in format AMSBIB
\Bibitem{Bar13}
\by N.~A.~Baraboshkina
\paper Approximation of harmonic functions by algebraic polynomials on a~circle of radius smaller than one with constraints on the unit circle
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2013
\vol 19
\issue 2
\pages 71--78
\mathnet{http://mi.mathnet.ru/timm933}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3363374}
\elib{http://elibrary.ru/item.asp?id=19053969}


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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. A. G. Babenko, T. Z. Naum, “One-sided integral approximations of the generalized Poisson kernel by trigonometric polynomials”, Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 38–48  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. G. Babenko, Yu. V. Kryakin, “Modified Bernstein function and a uniform approximation of some rational fractions by polynomials”, Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 45–59  mathnet  crossref  crossref  isi  elib
    3. O. L. Vinogradov, “Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions”, Siberian Math. J., 58:2 (2017), 190–204  mathnet  crossref  crossref  isi  elib  elib
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