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Extremal properties of the derivatives of the Newman polynomials

Authors:	Tamá s Erdé lyi

Institution:	Department of Mathematics, Texas A&M University, College Station, Texas 77843

Abstract:	Let $\Lambda _{n-1} := \{\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}\}$ be a set of 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 , , 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 $0 < q \leq p \leq \infty$ .

Keywords:	M\"{u}ntz polynomials exponential sums Markov-type inequality Nikolskii-type inequality Newman's inequality

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