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Local behaviour of polynomials

Authors:	D P Dryanov M A Qazi Q I Rahman

Institution:	Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada ; Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088 ; Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

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.

Keywords:	Trigonometric polynomials algebraic polynomials M Riesz's interpolation formula Schur's inequality Bernstein's inequality

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