Approximation theorems for uniformly continuous functions

Hager, Anthony W.

doi:10.1090/S0002-9947-1978-0510848-3

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.

References

Á. Császár, Gleichmässige Approximation und gleichmässige Stetigkeit, Acta Math. Acad. Sci. Hungar. 20 (1969), 253–261 (German). MR 253277, DOI 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 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 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), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 172–187. MR 0362236
J. R. Isbell, Algebras of uniformly continuous functions, Ann. of Math. (2) 68 (1958), 96–125. MR 103407, DOI 10.2307/1970045
J. R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR 0170323, DOI 10.1090/surv/012
M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85–91. MR 50264, DOI 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 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