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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Generalizations of Müntz’s Theorem via a Remez-type inequality for Müntz spaces
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by Peter Borwein and Tamás Erdélyi PDF
J. Amer. Math. Soc. 10 (1997), 327-349

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.
References
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Additional Information
  • Peter Borwein
  • Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
  • Email: pborwein@cecm.sfu.ca
  • Tamás Erdélyi
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Email: terdelyi@math.tamu.edu
  • 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
  • © Copyright 1997 by the authors
  • Journal: J. Amer. Math. Soc. 10 (1997), 327-349
  • MSC (1991): Primary 41A17; Secondary 30B10, 26D15
  • DOI: https://doi.org/10.1090/S0894-0347-97-00225-7
  • MathSciNet review: 1415318