Concerning the bounded case of the Bernstein-Nachbin approximation problem

Silvio MACHADO; João B. PROLLA

doi:10.2969/jmsj/02930451

Silvio MACHADO, João B. PROLLA

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JOURNAL FREE ACCESS

1977 Volume 29 Issue 3 Pages 451-458

DOI https://doi.org/10.2969/jmsj/02930451

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Published: 1977 Received: May 14, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Silvio MACHADO¹⁾, Joao B. PROLLA²⁾
Right : Silvio MACHADO¹⁾, João B. PROLLA²⁾

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. C. Buck, Approximation properties of vector valued functions, Pacific J. Math., 53 (1974), 85-94.
2) F. Cunningham, Jr., and N. M., Roy, Extreme functionals on an upper semicontinuous function space, Proc. Amer. Math. Soc., 42 (1974), 461-465.
3) I. Glicksberg, Measures orthogonal to algebras and sets of anti-symmetry, Trans. Amer. Math. Soc., 105 (1962), 415-435.
4) S. Machado, On Bishop's generalization of the Weierstrass-Stone theorem, to appear in Indag. Math.
5) S. Machado and J. B. Prolla, The general complex case of the Bernstein-Nachbin approximation problem, to appear in Ann. Inst. Fourier (Grenoble).
6) L. Nachbin, Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases, Ann. of Math., 81 (1965), 289-302.
7) L. Nachbin, S. Machado and J. B. Prolla, Concerning weighted approximation, vector fibrations, and algebras of operators, J. Approximation Theory, 6 (1972), 80-89.

Right : [1] R. C. Buck, Approximation properties of vector valued functions, Pacific J. Math., 53 (1974), 85-94.
[2] F. Cunningham, Jr., and N. M., Roy, Extreme functionals on an upper semicontinuous function space, Proc. Amer. Math. Soc., 42 (1974), 461-465.
[3] I. Glicksberg, Measures orthogonal to algebras and sets of anti-symmetry, Trans. Amer. Math. Soc., 105 (1962), 415-435.
[4] S. Machado, On Bishop's generalization of the Weierstrass-Stone theorem, to appear in Indag. Math.
[5] S. Machado and J. B. Prolla, The general complex case of the Bernstein-Nachbin approximation problem, to appear in Ann. Inst. Fourier (Grenoble).
[6] L. Nachbin, Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases, Ann. of Math., 81 (1965), 289-302.
[7] L. Nachbin, S. Machado and J. B. Prolla, Concerning weighted approximation, vector fibrations, and algebras of operators, J. Approximation Theory, 6 (1972), 80-89.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

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 Bishop's 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 Zorn's Lemma.

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