Lower bounds for derivatives of polynomials and Remez type inequalities
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- by Tamás Erdélyi and Paul Nevai PDF
- Trans. Amer. Math. Soc. 349 (1997), 4953-4972 Request permission
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.References
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Additional Information
- Tamás Erdélyi
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: terdelyi@math.tamu.edu
- Paul Nevai
- Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174
- Email: nevai@math.ohio-state.edu
- 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.).
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 4953-4972
- MSC (1991): Primary 33A65; Secondary 26C05, 42C05
- DOI: https://doi.org/10.1090/S0002-9947-97-01875-8
- MathSciNet review: 1407486