Generalizations of Müntz’s Theorem via a Remez-type inequality for Müntz spaces

Abstract

The principal result of this paper is a Remez-type inequality for Müntz polynomials: \begin{equation*}p(x) := \sum ^{n}_{i=0} a_{i} x^{\lambda _{i}}, \end{equation*} or equivalently for Dirichlet sums: \begin{equation*}P(t) := \sum ^{n}_{i=0}{a_{i} e^{-\lambda _{i} t}} ,\end{equation*} where $0 = \lambda _{0} < \lambda _{1} < \lambda _{2} <\cdots$. The most useful form of this inequality states that for every sequence $(\lambda _{i})^{\infty }_{i=0}$ satisfying $\sum ^{\infty }_{i=1} 1/\lambda _{i} < \infty$, there is a constant $c$ depending only on $\Lambda : = (\lambda _{i})^{\infty }_{i=0}$ and $s$ (and not on $n$, $\varrho$, or $A$) so that \begin{equation*}\|p\|_{[0, \varrho ]} \leq c \|p\|_{A}\end{equation*} for every Müntz polynomial $p$, as above, associated with $(\lambda _{i})^{\infty }_{i=0}$, and for every set $A \subset [\varrho ,1]$ of Lebesgue measure at least $s > 0$. Here $\|\cdot \|_{A}$ denotes the supremum norm on $A$. This Remez-type inequality allows us to resolve two reasonably long-standing conjectures. The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if $\sum ^{\infty }_{i=1} 1/\lambda _{i} < \infty$, then the set of products $\{ p_{1} p_{2} : p_{1}, p_{2} \in \text {span} \{x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots \}\}$ is not dense in $C[0,1]$. The second is a complete extension of Müntz’s classical theorem on the denseness of Müntz spaces in $C[0,1]$ to denseness in $C(A)$, where $A \subset [0,\infty )$ is an arbitrary compact set with positive Lebesgue measure. That is, for an arbitrary compact set $A \subset [0,\infty )$ with positive Lebesgue measure, $\text {span} \{ x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots \}$ is dense in $C(A)$ if and only if $\sum ^{\infty }_{i=1} 1/\lambda _{i} =\infty$. Several other interesting consequences are also presented.