Distribution of Points of Interpolation and of Zeros of Exactly Maximally Convergent Multipoint Padé Approximants

DOI: 10.4236/am.2015.65069   PDF   HTML     2,869 Downloads   3,180 Views  

Abstract

Given a regular compact set E in , a unit measure μ supported by , a triangular point set , and a function f , holomorphic on E , let πβ,fn,m be the associated multipoint β-Padé approximant of order (n,m) . We show that if the sequence πβ,fn,m , n∈Λ , n,k are uniformly distributed on with respect to u as n∈Λ . Furthermore, a result about the behavior of the zeros of the exact maximally convergent sequence Λ is provided, under the condition that Λ is “dense enough”.

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Kovacheva, R. (2015) Distribution of Points of Interpolation and of Zeros of Exactly Maximally Convergent Multipoint Padé Approximants. Applied Mathematics, 6, 737-744. doi: 10.4236/am.2015.65069.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Saff, E.B. and Totik, V. (1997) Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenschaften, 316.
http://dx.doi.org/10.1007/978-3-662-03329-6
[2] Carleson, L. (1964) Mergelyan’s Theorem on Uniform Polynomial Approximation. Mathematica Scandinavica, 15, 167-175.
[3] Saff, E.B. (1972) An Extension of Montessus de Ballore Theorem on the Convergence of Interpolation Rational Functions. Journal of Approximation Theory, 6, 63-67.
http://dx.doi.org/10.1016/0021-9045(72)90081-0
[4] Kovacheva, R.K. (1989) Generalized Padé Approximants of Kakehashi’s Type and Meromorphic Continuation of Functions. Deformation of Mathematical Structures, 151-159.
http://dx.doi.org/10.1007/978-94-009-2643-1_14
[5] Perron, O. (1929) Die Lehre von den Kettenbrüchen. Teubner, Leipzig.
[6] Gonchar, A.A. (1975) On the Convergence of Generalized Padé Approximants of Meromorphic Functions. Matematicheskii Sbornik, 98, 564-577. English Translation in Mathematics of the USSR-Sbornik, 27, 503-514.
[7] Tsuji, M. (1959) Potential Theory in Modern Function Theory. Maruzen, Tokyo.
[8] Bello Hernándes, M. and De la Calli Ysern, B. (2013) Meromorphic Continuation of Functions and Arbitrary Distribution of Interpolation Points. Journal of Mathematical Analysis and Applications, 403, 107-119.
http://dx.doi.org/10.1016/j.jmaa.2013.02.014
[9] Walsh, J.L. (1946) Overconvergence, Degree of Convergence, and Zeros of Sequences of Analytic Functions. Duke Mathematical Journal, 13, 195-234.
http://dx.doi.org/10.1215/S0012-7094-46-01320-8
[10] Walsh, J.L. (1959) The Analogue for Maximally Convergent Polynomials of Jentzsch’s Theorem. Duke Mathematical Journal, 26, 605-616.
http://dx.doi.org/10.1215/S0012-7094-59-02658-4
[11] Walsh, J.L. (1969) Interpolation and Approximation by Rational Functions in the Complex Domain. Vol. 20, American Mathematical Society Colloquium Publications, New York.
[12] Ikonomov, N. (2013) Multipoint Padé Approximants and Uniform Distribution of Points. Comptes Rendus de l’Aca- demie Bulgare des Sciences, 66, 1097-1105.
[13] Ikonomov, N. (2014) Generalized Padé Approximants for Plane Condenser. Mathematica Slovaca, Springer, Accepted for Publication in 2014.
[14] Grothmann, R. (1996) Distribution of Interpolation Points. Arkiv f?r matematik, 34, 103-117.
http://dx.doi.org/10.1007/BF02559510
[15] Ikonomov, N. and Kovacheva, R.K. (2014) Distribution of Points of Interpolation of Multipoint Padé Approximants. AIP Conference Proceedings, AMEE2014, 1631, 292-296.
http://dx.doi.org/10.1063/1.4902489
[16] Blatt, H.P. and Kovacheva, R.K. (2015) Distribution of Interpolation Points of Maximally Convergent Multipoint Padé Approximants. Journal of Approximation Theory, 191, 46-57.
[17] Kovacheva, R.K. (2010) Normal Families of Meromorphic Functions. Comptes Rendus de l'Academie Bulgare des Sciences, 63, 807-814.

  
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