Extremal properties of the derivatives of the Newman polynomials

Abstract:

Let $\Lambda _{n-1} := \{\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}\}$ be a set of $n$ distinct positive numbers. The span of \begin{equation*}\{e^{-\lambda _{1}t}, e^{-\lambda _{2}t}, \ldots , e^{-\lambda _{n}t}\} \end{equation*} over ${\mathbb {R}}$ will be denoted by \begin{equation*} E(\Lambda _{n-1}) := \operatorname {span} \{e^{-\lambda _{1}t}, e^{-\lambda _{2}t}, \ldots , e^{-\lambda _{n}t}\}\,. \end{equation*} 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 )^{\mu + \frac {1}{q} - \frac {1}{p}} &\qquad \leq \sup _{Q \in E(\Lambda _{n-1})} {\frac {\|Q^{(\mu )}\|_{L_{p}[0,\infty )}}{\|Q\|_{L_{q}[0,\infty )}}} \leq c_{2}(p,q,\mu ) \left (\sum _{j=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 $0 < q \leq p \leq \infty$.