Muntz's Theorem on compact subsets of positive measure

Borwein, Peter and Erdelyi, Tamas (1995) Muntz's Theorem on compact subsets of positive measure. [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 $(\lambda_i)_{i=0}^{\infty}$ is a sequence of distinct real numbers. The most useful form of this inequality states that for every sequence $(\lambda_i)^\infty_{i=0}$ satisfying $$\sum^\infty_{\scriptstyle{i = 0} \atop \scriptstyle{\lambda_i \neq 0}}\frac{1}{|\lambda_i|} < \infty$$ there is a constant $c$ depending only on $(\lambda_i)^\infty_{i=0}$, $A$, $\alpha$, and $\beta$ (and not on $n$ or $A$) so that the inequality $$\|p\|_{[\alpha, \beta]} \leq c \, \|p\|_A$$ holds for every M\"untz polynomial $p$, as above, associated with $(\lambda_i)^\infty_{i=0}$, for every set $A \subset [0,\infty)$ of positive Lebessgue measure, and for every $$[\alpha,\beta] \subset (\text{ess inf }A,\text{ess sup }A).$$ Here $\|\cdot \|_A$ denotes the supremum norm on $A$. This Remez-type inequality allows us to resolve several problems. Most notably we show that the M\"untz-type theorems of Clarkson, Erd\H os, and Schwartz on the denseness of $$\text{span}\{x^{\lambda_0}, x^{\lambda_1}, \ldots \}, \qquad \lambda_i \in {\Bbb R} \enskip \text{distinct}$$ on $[a,b]$, $a > 0$, remain valid with $[a,b]$ replaced by an arbitrary compact set $A \subset (0,\infty)$ of positive Lebesgue measure. This extends earlier results of the authors under the assumption that the numbers $\lambda_i$ are nonnegative.