Concerning the bounded case of the Bernstein-Nachbin approximation problem

概要

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Recently we gave a solution to the Bernstein-Nachbin Approximation Problem in the general complex case. As a corollary, we obtained the quasi-analytic, the analytic, and the bounded criteria for localizability in the general complex case. This generalizes the known results of the real or self-adjoint complex cases, in the same way that Bishops Theorem generalizes the Weierstrass-Stone Theorem. In this paper, we present a direct proof of the bounded criterion for localizability, and show how it can be used to get a new proof of our solution of the Bernstein-Nachbin Approximation Problem. The proof in the real case is based in the idea of the proof of the Weierstrass-Stone Theorem discovered by one of us; the general complex case follows by Zorns Lemma.
社団法人日本数学会の論文