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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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On $P$-orderings, rings of integer-valued polynomials, and ultrametric analysis
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by Manjul Bhargava PDF
J. Amer. Math. Soc. 22 (2009), 963-993 Request permission

Abstract:

We introduce two new notions of “$P$-ordering” and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of $P$-orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) $P$-adic analysis.

Specifically, we first use these notions of $P$-orderings and factorials to construct explicit Pólya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain.

Second, we classify “smooth” functions on an arbitrary compact subset $S$ of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on $S$ satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler’s Theorem (classifying the functions that are continuous on $\mathbb {Z}_p$) to a very general setting. In particular, our constructions prove that, for any $\epsilon >0$, the functions in any of the above Banach spaces can be $\epsilon$-approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallée-Poussin, and Bernstein. Our proofs are effective.

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Additional Information
  • Manjul Bhargava
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • MR Author ID: 623882
  • Email: bhargava@math.princeton.edu
  • Received by editor(s): February 25, 2008
  • Published electronically: May 27, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 963-993
  • MSC (2000): Primary 11C08, 11S80; Secondary 13F20, 13B25
  • DOI: https://doi.org/10.1090/S0894-0347-09-00638-9
  • MathSciNet review: 2525777