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Mat. Sb., 2002, Volume 193, Number 1, Pages 3–72 (Mi msb619)  

This article is cited in 54 scientific papers (total in 55 papers)

Sharp constants for rational approximations of analytic functions

A. I. Aptekarev

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: Theorems describing the sharp constants for the approximation of a general class of analytic functions by rational functions are proved. Magnus's conjecture on the sharp constant for the approximation of $e^{-z}$ on $[0,\infty]$ is established as a consequence. For the proof of the theorems new formulae expressing the strong asymptotics of polynomials orthogonal with respect to a varying complex weight are obtained.


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Sbornik: Mathematics, 2002, 193:1, 1–72

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UDC: 517.53
MSC: 30E10, 41A20
Received: 12.03.2001

Citation: A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Mat. Sb., 193:1 (2002), 3–72; Sb. Math., 193:1 (2002), 1–72

Citation in format AMSBIB
\by A.~I.~Aptekarev
\paper Sharp constants for rational approximations of analytic functions
\jour Mat. Sb.
\yr 2002
\vol 193
\issue 1
\pages 3--72
\jour Sb. Math.
\yr 2002
\vol 193
\issue 1
\pages 1--72

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    2. Kuijlaars A.B.J., Martínez-Finkelshtein A., “Strong asymptotics for Jacobi polynomials with varying nonstandard parameters”, J. Anal. Math., 94:1 (2004), 195–234  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. Kuijlaars A.B.J., McLaughlin K.T.R., “Asymptotic zero behavior of Laguerre polynomials with negative parameter”, Constr. Approx., 20:4 (2004), 497–523  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    4. Wielonsky F., “Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function”, J. Approx. Theory, 131:1 (2004), 100–148  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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    7. McLaughlin K.T.-R., Vartanian A.H., Zhou X., “Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights”, IMRP Int. Math. Res. Pap., 2006, 62815, 216 pp.  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    15. Baratchart L., Yattselev M., “Convergent interpolation to Cauchy integrals over analytic arcs”, Found. Comput. Math., 9:6 (2009), 675–715  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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    22. Baratchart L., Yattselev M., “Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights”, Internat. Math. Res. Notices, 2010, no. 22, 4211–4275  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    23. Stahl H., “A potential-theoretic problem connected with complex orthogonality”, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., 507, Amer. Math. Soc., Providence, RI, 2010, 255–285  crossref  mathscinet  isi
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