# The general complex case of the Bernstein-Nachbin approximation problem

Annales de l'institut Fourier (1978)

- Volume: 28, Issue: 1, page 193-206
- ISSN: 0373-0956

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topMachado, S., and Prolla, Joao Bosco. "The general complex case of the Bernstein-Nachbin approximation problem." Annales de l'institut Fourier 28.1 (1978): 193-206. <http://eudml.org/doc/74345>.

@article{Machado1978,

abstract = {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.},

author = {Machado, S., Prolla, Joao Bosco},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {1},

pages = {193-206},

publisher = {Association des Annales de l'Institut Fourier},

title = {The general complex case of the Bernstein-Nachbin approximation problem},

url = {http://eudml.org/doc/74345},

volume = {28},

year = {1978},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 28

IS - 1

SP - 193

EP - 206

AB - 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.

LA - eng

UR - http://eudml.org/doc/74345

ER -

## References

top- [1] E. BISHOP, A generalization of the Stone-Weierstrass theorem, Pacific J. Math., 11 (1961), 777-783. Zbl0104.09002MR24 #A3502
- [2] I. GLICKSBERG, Measures orthogonal to algebras and sets of antisymmetry, Trans, Amer. Math. Soc., 105 (1962), 415-435. Zbl0111.11801MR30 #4164
- [3] R.I. JEWETT, A variation on the Stone-Weierstrass theorem, Proc. Amer. Math. Soc., 14 (1963), 690-693. Zbl0122.35004MR27 #2854
- [4] G. KLEINSTUCK, Der beschränkte Fall des gewichteten Approximationsproblems für vektorwertige Funktionen, Manuscripta Math., 17 (1975), 123-149. Zbl0343.41033MR53 #11364
- [5] L. NACHBIN, On the priority of algebras of continuous functions in weighted approximation, to appear in Symposia Mathematica. Zbl0338.41034
- [6] L. NACHBIN, Elements of Approximation Theory, D. van Nostran Co., Inc., 1967. Reprinted by R. Krieger Co., Inc., 1976. Zbl0173.41403
- [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. Zbl0238.46041MR45 #2474
- [8] J.B. PROLLA, Bishop's generalized Stone-Weierstrass theorem for weighted spaces, Math. Ann., 191 (1971), 283-289. Zbl0202.12603MR44 #7200
- [9] W. RUDIN, Real and complex analysis, McGraw-Hill, New York, 1966. Zbl0142.01701MR35 #1420
- [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. Zbl0321.41029MR52 #6283

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