The general complex case of the Bernstein-Nachbin approximation problem
Annales de l'Institut Fourier, Volume 28 (1978) no. 1, p. 193-206
We present a solution to the (strict) Bernstein-Nachbin approximation problem in the general complex case. As a corollary, we get proofs of the analytic, the quasi-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. However, even in the real or self-adjoint complex cases, the results that we obtain are stronger than the previously known results of the literature.
On présente ici une solution du problème d’approximation de Bernstein-Nachbin dans le cas complexe général, c’est-à-dire non nécessairement auto-adjointe. On généralise ainsi les résultats connus de cette théorie de la même façon que le théorème d’approximation de Bishop généralise le théorème de Weierstrass-Stone.
@article{AIF_1978__28_1_193_0,
author = {Machado, S. and Prolla, Joao Bosco},
title = {The general complex case of the Bernstein-Nachbin approximation problem},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {28},
number = {1},
year = {1978},
pages = {193-206},
doi = {10.5802/aif.685},
zbl = {0365.41007},
mrnumber = {81g:46069},
language = {en},
url = {http://www.numdam.org/item/AIF_1978__28_1_193_0}
}

Machado, S.; Prolla, Joao Bosco. The general complex case of the Bernstein-Nachbin approximation problem. Annales de l'Institut Fourier, Volume 28 (1978) no. 1, pp. 193-206. doi : 10.5802/aif.685. http://www.numdam.org/item/AIF_1978__28_1_193_0/

[1] E. Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math., 11 (1961), 777-783. | MR 24 #A3502 | Zbl 0104.09002

[2] I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans, Amer. Math. Soc., 105 (1962), 415-435. | MR 30 #4164 | Zbl 0111.11801

[3] R.I. Jewett, A variation on the Stone-Weierstrass theorem, Proc. Amer. Math. Soc., 14 (1963), 690-693. | MR 27 #2854 | Zbl 0122.35004

[4] G. Kleinstuck, Der beschränkte Fall des gewichteten Approximationsproblems für vektorwertige Funktionen, Manuscripta Math., 17 (1975), 123-149. | MR 53 #11364 | Zbl 0343.41033

[5] L. Nachbin, On the priority of algebras of continuous functions in weighted approximation, to appear in Symposia Mathematica. | Zbl 0338.41034

[6] L. Nachbin, Elements of Approximation Theory, D. van Nostran Co., Inc., 1967. Reprinted by R. Krieger Co., Inc., 1976. | Zbl 0173.41403

[7] L. Nachbin, S. Machado, and J.B. Prolla, Weighted approximation, vector fibrations, and algebras of operators, Journal Math. Pures et appl., 50 (1971), 299-323. | MR 45 #2474 | Zbl 0238.46041

[8] J.B. Prolla, Bishop's generalized Stone-Weierstrass theorem for weighted spaces, Math. Ann., 191 (1971), 283-289. | MR 44 #7200 | Zbl 0202.12603

[9] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. | MR 35 #1420 | Zbl 0142.01701

[10] W.H. Summers, Weighted approximation for modules of continuous functions II, in “Analyse fonctionnelle et applications” (Editor L. Nachbin), Hermann, Paris, 1975, p. 277-283. | MR 52 #6283 | Zbl 0321.41029