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- W1978170363 abstract "In 1941 Seidel and Walsh [3] proved the existence of an entire function F of a complex variable such that every function analytic in a simply connected region of the complex plane is the uniform limit on compact sets of a sequence of translates of F. This result generalizes a theorem of G. D. Birkhoff [1] on entire functions. In the present note an analogous theorem is proved for continuous real or complex functions on a more general class of topological spaces where the role of polynomial approximation in the above proofs is assumed by a sequence of functions constructed using Urysohn's lemma. Let X be a locally compact hausdorff space with the following properties: there exist countable sequences { Cn } and {Io-, } of disjoint compact sets and homeomorphisms of X onto itself, respectively, such that for every compact K, Ia. KnCn=0 and lb. KCCno-a,, except for finitely many n. Such an X is evidently not compact but is countable at infinity, since each point lies in some Cnon. Thus the compact open topology on the algebra 2t of all continuous real or complex valued functions on X is the topology of sequential convergence in a suitable Frechet metric on W." @default.
- W1978170363 created "2016-06-24" @default.
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- W1978170363 date "1957-06-01" @default.
- W1978170363 modified "2023-09-26" @default.
- W1978170363 title "Approximation by the translates of a single function" @default.
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- W1978170363 doi "https://doi.org/10.1090/s0002-9939-1957-0096061-7" @default.
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