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

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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UDC: 517.51

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
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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