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On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation

Published online by Cambridge University Press:  20 November 2018

K. G. Ivanov ,
E. B. Saff  and
V. Totik
K. G. Ivanov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Science, Sofia, 1090 Bulgaria
E. B. Saff
Affiliation:
Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620, USA
V. Totik
Affiliation:
Bolyai Institute, Aradi V. tere 1, Szeged, 6720 Hungary Department of Mathematics, University of South Florida, Tampa, Florida 33620, USA
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Abstract

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Assume ƒ is continuous on the closed disk D1 : |z| ≤ 1, analytic in |z| ≤ 1, but not analytic on D1. Our concern is with the behavior of the zeros of the polynomials of best uniform approximation to ƒ on D1. It is known that, for such ƒ, every point of the circle |z| = 1 is a cluster point of the set of all zeros of Here we show that this property need not hold for every subsequence of the Specifically, there exists such an f for which the zeros of a suitable subsequence all tend to infinity. Further, for near-best polynomial approximants, we show that this behavior can occur for the whole sequence. Our examples can be modified to apply to approximation in the Lq-norm on |z|= 1 and to uniform approximation on general planar sets (including real intervals).

Keywords

41A20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 5 , 01 October 1991 , pp. 1010 - 1021
Copyright
Copyright © Canadian Mathematical Society 1991

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