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

Authors: Peter Borwein and Tamás Erdélyi
Journal: J. Amer. Math. Soc. 10 (1997), 327-349
MSC (1991): Primary 41A17; Secondary 30B10, 26D15
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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Remez inequality, M\"{u}ntz's Theorem, M\"{u}ntz spaces, Dirichlet sums, density
Received by editor(s): June 10, 1994
Received by editor(s) in revised form: September 20, 1996
Additional Notes: Research of the first author was supported, in part, by NSERC of Canada. Research of the second author was supported, in part, by NSF under Grant No. DMS-9024901 and conducted while an NSERC International Fellow at Simon Fraser University.
Dedicated: Dedicated to the memory of Paul Erdős
Article copyright: © Copyright 1997 by the authors

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