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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Approximation theorems for uniformly continuous functions
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by Anthony W. Hager PDF
Trans. Amer. Math. Soc. 236 (1978), 263-273 Request permission

Abstract:

Let X be a set, A a family of real-valued functions on X which contains the constants, ${\mu _A}$ the weak uniformity generated by A, and $U({\mu _A}X)$ the collection of uniformly continuous functions to the real line R. The problem is how to construct $U({\mu _A}X)$ from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely A-equiuniform, order-one subsets of ${A^ + }$ is uniformly dense in $U({\mu _A}X)$. Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly locally-A (resp., uniformly locally piecewise-A) is uniformly dense in $U({\mu _A}X)$. Further, for any A, a finitely A-uniform function is just a composition $F \circ ({a_1}, \ldots ,{a_p})$ for some ${a_1}, \ldots ,{a_p} \in A$ and F uniformly continuous on the range of $({a_1}, \ldots ,{a_p})$ in ${R^p}$. Thus, such compositions are dense in $U({\mu _A}X)$. For $BU({\mu _A}X)$, the compositions with $F \in BU({R^p})$ are dense (B denoting bounded functions). So, in a sense, to know $U({\mu _A}X)$ it suffices to know A and subspaces of the spaces ${R^p}$, and to know $BU({\mu _A}X)$, A and the spaces ${R^p}$ suffice.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 236 (1978), 263-273
  • MSC: Primary 41A65; Secondary 41A30
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0510848-3
  • MathSciNet review: 0510848