no. 1
The general complex case of the Bernstein-Nachbin approximation problem
Annales de l'Institut Fourier, Tome 28 (1978) no. 1, pp. 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},
     pages = {193--206},
     year = {1978},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {1},
     doi = {10.5802/aif.685},
     zbl = {0365.41007},
     mrnumber = {81g:46069},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.685/}
}
TY  - JOUR
AU  - Machado, S.
AU  - Prolla, Joao Bosco
TI  - The general complex case of the Bernstein-Nachbin approximation problem
JO  - Annales de l'Institut Fourier
PY  - 1978
SP  - 193
EP  - 206
VL  - 28
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.685/
DO  - 10.5802/aif.685
LA  - en
ID  - AIF_1978__28_1_193_0
ER  - 
%0 Journal Article
%A Machado, S.
%A Prolla, Joao Bosco
%T The general complex case of the Bernstein-Nachbin approximation problem
%J Annales de l'Institut Fourier
%D 1978
%P 193-206
%V 28
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.685/
%R 10.5802/aif.685
%G en
%F 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, Tome 28 (1978) no. 1, pp. 193-206. doi: 10.5802/aif.685

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

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

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

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

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

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

Cité par Sources :