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Mat. Sb., 1992, Volume 183, Number 6, Pages 97–110 (Mi msb1048)  

This article is cited in 1 scientific paper (total in 1 paper)

$L_p$ extensions of Gonchar's inequality for rational functions

A. L. Levin, E. B. Saff


Abstract: Given a condenser $ (E,  F)$ in the complex plane, let $ C(E,  F)$ denote its capacity and let $ \mu^*=\mu_E^*-\mu_F^*$ be the (signed) equilibrium distribution for $ (E,  F)$. Given a finite positive measure $\mu$ on $E\cup F$, let
$$ G(\mu_E')=\exp( \int\log(d\mu/d\mu_E^*) d\mu_E^*),\quad G(\mu_F')=\exp( \int\log(d\mu/d\mu_F^*) d\mu_F^*). $$
We show that for $0<p,q<\infty$ and for any rational function $r_n$ of order $n$
\begin{equation} \|r_n\|_{L_p(d\mu,E)}\|1/r_n\|_{L_q(d\mu,F)}\ge e^{-n/C(E,F)}G^{1/p}(\mu_E') G^{1/q}(\mu_E'), \tag{1} \end{equation}
which extends a classical result due to A. A. Gonchar. For a symmetric condenser we also obtain a sharp lower bound for $\|r_n-\lambda\|_{L_p(d\mu, E\cup F)}$, where $\lambda=\lambda(z)$ is equal to $0$ on $E$ and $1$ on $F$. The question of exactness of (1) and the relation to certain $n$-widths are also discussed.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 76:1, 199–210

Bibliographic databases:

UDC: 517.5
MSC: Primary 30A10, 30C85; Secondary 31A15
Received: 12.06.1991

Citation: A. L. Levin, E. B. Saff, “$L_p$ extensions of Gonchar's inequality for rational functions”, Mat. Sb., 183:6 (1992), 97–110; Russian Acad. Sci. Sb. Math., 76:1 (1993), 199–210

Citation in format AMSBIB
\Bibitem{LevSaf92}
\by A.~L.~Levin, E.~B.~Saff
\paper $L_p$ extensions of Gonchar's inequality for rational functions
\jour Mat. Sb.
\yr 1992
\vol 183
\issue 6
\pages 97--110
\mathnet{http://mi.mathnet.ru/msb1048}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1186780}
\zmath{https://zbmath.org/?q=an:0782.30030|0766.30033}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1993SbMat..76..199L}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1993
\vol 76
\issue 1
\pages 199--210
\crossref{https://doi.org/10.1070/SM1993v076n01ABEH003408}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1993MD58900011}


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    This publication is cited in the following articles:
    1. M.-P. Istace, J.-P. Thiran, “On the Third and Fourth Zolotarev Problems in the Complex Plane”, SIAM J Numer Anal, 32:1 (1995), 249  crossref  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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