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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant


Authors: Q. I. Rahman and G. Schmeisser
Journal: Trans. Amer. Math. Soc. 310 (1988), 693-702
MSC: Primary 26D05
DOI: https://doi.org/10.1090/S0002-9947-1988-0946426-8
MathSciNet review: 946426
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Abstract: If $ p$ is a polynomial of degree at most $ n$ such that $ \vert p(x)\vert \leqslant \sqrt {1 - {x^2}} $ for $ - 1 \leqslant x \leqslant 1$, then for each $ k$, $ \max \vert{p^{(k)}}(x)\vert$ on $ [ - 1,\,1]$ is maximized by the polynomial $ ({x^2} - 1){U_{n - 2}}(x)$ where $ {U_m}$ is the $ m$th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume $ \vert p(x)\vert \leqslant \sqrt {1 - {x^2}} $ at some appropriately chosen set of $ n + 1$ points in $ [ - 1,\,1]$. The problem is inspired by a well-known extension of Markov's inequality due to Duffin and Schaeffer.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0946426-8
Keywords: Markov's inequality, Chebyshev polynomial of the first kind, interpolation, circular majorant, Chebyshev polynomial of the second kind
Article copyright: © Copyright 1988 American Mathematical Society