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Lower bounds for derivatives of polynomials and Remez type inequalities

Authors: Tamás Erdélyi and Paul Nevai
Journal: Trans. Amer. Math. Soc. 349 (1997), 4953-4972
MSC (1991): Primary 33A65; Secondary 26C05, 42C05
MathSciNet review: 1407486
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Abstract: P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89-95] proved that if all the zeros of a polynomial $p$ lie in the unit interval $I% \overset {\text {def}}{=} [-1,1]$, then $\|p'\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\;$. Our goal is to study the feasibility of $\lim _{{n\to \infty }% }{\|p_{n}'\|_{X}} / {\|p_{n}\|_{Y}} =\infty $ for sequences of polynomials $\{p_{n}\}_{n\in \mathbb N% }$ whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.

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Additional Information

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Paul Nevai
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174

Keywords: Markov type inequalities, Remez type inequalities, Tur\'{a}n type inequalities, derivatives, algebraic polynomials, trigonometric polynomials, generalized polynomials
Received by editor(s): April 20, 1996
Additional Notes: This material is based upon work supported by the National Science Foundation under Grants No. DMS–9024901 (both authors) and No. DMS–940577 (P. N.).
Article copyright: © Copyright 1997 American Mathematical Society

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