Approximation theorems for uniformly continuous functions

Author:
Anthony W. Hager

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

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Abstract | References | Similar Articles | Additional Information

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.

- Á. Császár,
*Gleichmässige Approximation und gleichmässige Stetigkeit*, Acta Math. Acad. Sci. Hungar.**20**(1969), 253–261 (German). MR**253277**, DOI https://doi.org/10.1007/BF01894894 - Jens Erik Fenstad,
*On $l$-groups of uniformly continuous functions. I. Approximation theory*, Math. Z.**82**(1963), 434–444. MR**159215**, DOI https://doi.org/10.1007/BF01111541 - Jens Erik Fenstad,
*On $l$-groups of uniformly continuous functions. II. Representation theory*, Math. Z.**83**(1964), 46–56. MR**171262**, DOI https://doi.org/10.1007/BF01111107 - Anthony W. Hager,
*Vector lattices of uniformly continuous functions and some categorical methods in uniform spaces*, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf. Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Springer, Berlin, 1974, pp. 172–187. Lecture Notes in Math., Vol. 378. MR**0362236** - J. R. Isbell,
*Algebras of uniformly continuous functions*, Ann. of Math. (2)**68**(1958), 96–125. MR**103407**, DOI https://doi.org/10.2307/1970045 - J. R. Isbell,
*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323** - M. Katětov,
*On real-valued functions in topological spaces*, Fund. Math.**38**(1951), 85–91. MR**50264**, DOI https://doi.org/10.4064/fm-38-1-85-91 - Wilhelm Maak,
*Eine Verallgemeinerung des Weierstrassschen Approximationssatzes*, Arch. Math. (Basel)**6**(1955), 188–193 (German). MR**69256**, DOI https://doi.org/10.1007/BF01900738 - Georg Nöbeling and Heinz Bauer,
*Allgemeine Approximationskriterien mit Anwendungen*, Jber. Deutsch. Math.-Verein.**58**(1955), no. Abt. 1, 54–72 (German). MR**74555**

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Keywords:
Stone-Weierstrass,
uniformly continuous,
uniform approximation

Article copyright:
© Copyright 1978
American Mathematical Society