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 Trudy Inst. Mat. i Mekh. UrO RAN, 2009, Volume 15, Number 1, Pages 66–78 (Mi timm205)

Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces $L^r$ ($r>1$).

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

Abstract: Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle $|z|=1$ with the weight $\varphi(\tau):=h(\tau)|\sin(\tau/2)|^{-1}g(|\sin(\tau/2)|)$ ($\tau\in\mathbb R$), where $g(t)$ is a concave modulus of continuity slowly changing at zero such that $t^{-1}g(t)\in L^1[0,1]$ and $h(\tau)$ is a positive function from the class $C_{2\pi}$ with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point $t=1$ to the greatest zero of a polynomial orthogonal on the segment [-1,1].

Keywords: orthogonal polynomials, pointwise estimates, the Szegő function

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2009, 265, suppl. 1, S64–S77

Bibliographic databases:

UDC: 517.5

Citation: V. M. Badkov, “Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces $L^r$ ($r>1$).”, Trudy Inst. Mat. i Mekh. UrO RAN, 15, no. 1, 2009, 66–78; Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S64–S77

Citation in format AMSBIB
\paper Pointwise estimates of polynomials orthogonal on a~circle with respect to a~weight not belonging to the spaces $L^r$ ($r>1$).
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2009
\vol 15
\issue 1
\pages 66--78
\mathnet{http://mi.mathnet.ru/timm205}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2728956}
\elib{http://elibrary.ru/item.asp?id=11929778}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2009
\vol 265
\issue , suppl. 1
\pages S64--S77
\crossref{https://doi.org/10.1134/S0081543809060066}

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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. V. M. Badkov, “Some properties of Jacobi polynomials orthogonal on a circle”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S49–S58
2. V. M. Badkov, “Estimates of the Lebesgue function of Fourier sums over trigonometric polynomials orthogonal with a weight not belonging to the spaces $L^r$ $(r>1)$”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 21–32
3. Vasil'eva A.A., “Kolmogorov widths of weighted Sobolev classes on a domain for a special class of weights”, Russ. J. Math. Phys., 18:3 (2011), 353–385
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