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Local behaviour of polynomials

Authors: D. P. Dryanov, M. A. Qazi and Q. I. Rahman
Journal: Math. Comp. 73 (2004), 1345-1364
MSC (2000): Primary 42A05, 26D05, 26D10, 30C10, 30A10
Published electronically: July 28, 2003
MathSciNet review: 2047090
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the local behaviour of a trigonometric polynomial $t(\theta )\,:=\,\sum _{\nu =-n}^{n}\,a_{\nu }\,e^{{i}\nu \theta }$ around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in $[0 \,,\, 2 \pi )$. The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form $\sum _{\nu =0}^{n}\,a_{\nu }\,x^{\nu }$ are also proved.

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Additional Information

D. P. Dryanov
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

M. A. Qazi
Affiliation: Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088

Q. I. Rahman
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

Keywords: Trigonometric polynomials, algebraic polynomials, M. Riesz's interpolation formula, Schur's inequality, Bernstein's inequality
Received by editor(s): August 20, 2002
Received by editor(s) in revised form: December 22, 2002
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society