# Generalizations of Muntz's Theorem via a Remez-Type Inequality for Muntz Spaces

Borwein, Peter and Erdelyi, Tamas (1993) Generalizations of Muntz's Theorem via a Remez-Type Inequality for Muntz Spaces. [Preprint]

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The principal result of this paper is a Remez-type inequality for M\"untz polynomials: $$p(x) := \sum^n_{i=0} a_i x^{\lambda_i}$$ or equivalently for Dirichlet sums: $$P(t) := \sum^n_{i=0}a_i e^{-\lambda_i t}$$ 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 $$\|p\|_{[0, \varrho]} \leq c \|p\|_A$$ for every M\"untz 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, due to D. J. Newman and dating from 1978, 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 M\"untz's classical theorem on the denseness of M\"untz 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.