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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The full Markov-Newman inequality for Müntz polynomials on positive intervals


Authors: David Benko, Tamás Erdélyi and József Szabados
Journal: Proc. Amer. Math. Soc. 131 (2003), 2385-2391
MSC (2000): Primary 41A17; Secondary 30B10, 26D15
Published electronically: February 26, 2003
MathSciNet review: 1974635
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Abstract: For a function $f$ defined on an interval $[a,b]$ let

\begin{displaymath}\Vert f\Vert _{[a,b]} := \sup \{\vert f(x)\vert: x \in [a,b]\}\,. \end{displaymath}

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let $n \geq 1$ be an integer. Let $\lambda _{0}, \lambda _{1}, \ldots , \lambda _{n}$ be $n+1$ distinct real numbers. Let $0 < a < b$. Then

\begin{displaymath}\begin{split}\frac{1}{3} \sum _{j=0}^{n}{\vert\lambda _{j}\ve... ...lambda _{j}\vert} + \frac{128}{\log (b/a)}(n+1)^2\,,\end{split}\end{displaymath}

where the supremum is taken for all $Q \in \text{\rm span}\{x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots , x^{\lambda _{n}}\}$ (the span is the linear span over ${\mathbb{R}}$).


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Additional Information

David Benko
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: benko@math.tamu.edu

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: terdelyi@math.tamu.edu

József Szabados
Affiliation: Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest, Hungary, H-1364
Email: szabados@renyi.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06980-6
PII: S 0002-9939(03)06980-6
Keywords: M\"{u}ntz polynomials, exponential sums, Markov-type inequality, Newman's inequality
Received by editor(s): March 2, 2002
Published electronically: February 26, 2003
Additional Notes: The second author’s research was supported, in part, by the NSF under Grant No. DMS-0070826
The third author’s research was supported by OTKA Grant No. T32872
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society