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

 

Extremal properties of the derivatives of the Newman polynomials


Author: Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 131 (2003), 3129-3134
MSC (2000): Primary 41A17; Secondary 30B10, 26D15
Published electronically: February 28, 2003
MathSciNet review: 1992853
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Abstract: Let $\Lambda _{n-1} := \{\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}\}$be a set of $n$ distinct positive numbers. The span of

\begin{displaymath}\{e^{-\lambda _{1}t}, e^{-\lambda _{2}t}, \ldots , e^{-\lambda _{n}t}\} \end{displaymath}

over ${\mathbb{R}}$ will be denoted by

\begin{displaymath}E(\Lambda _{n-1}) := \text{\rm span}\{e^{-\lambda _{1}t}, e^{-\lambda _{2}t}, \ldots , e^{-\lambda _{n}t}\}\,. \end{displaymath}

Our main result of this note is the following.


Theorem. Suppose $0 < q \leq p \leq \infty $. Let $\mu $ be a non-negative integer. Then there are constants $c_{1}(p,q,\mu ) > 0$ and $c_{2}(p,q,\mu ) > 0$ depending only on $p$, $q$, and $\mu $ such that
\begin{align*}&c_{1}(p,q,\mu ) \left ( \sum _{j=1}^{n}{\lambda _{j}}\right )^{\m... ...=1}^{n}{\lambda _{j}}\right )^{\mu + \frac{1}{q} - \frac{1}{p}} \,, \end{align*}
where the lower bound holds for all $0 < q \leq p \leq \infty $ and for all $\mu \geq 0$, while the upper bound holds when $\mu = 0$ and $0 < q \leq p \leq \infty $ and when $\mu \geq 1$, $p \geq 1$, and $0 < q \leq p \leq \infty $.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06986-7
PII: S 0002-9939(03)06986-7
Keywords: M\"{u}ntz polynomials, exponential sums, Markov-type inequality, Nikolskii-type inequality, Newman's inequality
Received by editor(s): April 29, 2002
Published electronically: February 28, 2003
Additional Notes: This research was supported, in part, by the NSF under Grant No. DMS-0070826
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society