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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 961209, 21 pages
Research Article

On the Distribution of Zeros and Poles of Rational Approximants on Intervals

1Department of Mathematical Sciences, Kent State University, Kent, OH 44242-0001, USA
2Lehrstuhl für Mathematik-Angewandte Mathematik, Mathematisch-Geographische Fakulätt, Katholische Universität Eichstätt-Ingolstadt, 85071 Eichstätt, Germany
3Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Received 29 February 2012; Accepted 8 May 2012

Academic Editor: Karl Joachim Wirths

Copyright © 2012 V. V. Andrievskii et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The distribution of zeros and poles of best rational approximants is well understood for the functions 𝑓 ( 𝑥 ) = | 𝑥 | 𝛼 , 𝛼 > 0 . If 𝑓 𝐶 [ 1 , 1 ] is not holomorphic on [ 1 , 1 ] , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [ 1 , 1 ] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, 𝑎 -values, and poles of best real rational approximants of degree at most 𝑛 to a function 𝑓 𝐶 [ 1 , 1 ] that is real-valued, but not holomorphic on [ 1 , 1 ] . Generalizations to the lower half of the Walsh table are indicated.