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Transactions of the American Mathematical Society

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Approximation theorems for uniformly continuous functions

Author: Anthony W. Hager
Journal: Trans. Amer. Math. Soc. 236 (1978), 263-273
MSC: Primary 41A65; Secondary 41A30
MathSciNet review: 0510848
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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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Keywords: Stone-Weierstrass, uniformly continuous, uniform approximation
Article copyright: © Copyright 1978 American Mathematical Society