The general complex case of the Bernstein-Nachbin approximation problem

S. Machado; Joao Bosco Prolla

Annales de l'institut Fourier (1978)

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

Abstract

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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.

How to cite

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Machado, 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

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  1. [1] E. BISHOP, A generalization of the Stone-Weierstrass theorem, Pacific J. Math., 11 (1961), 777-783. Zbl0104.09002MR24 #A3502
  2. [2] I. GLICKSBERG, Measures orthogonal to algebras and sets of antisymmetry, Trans, Amer. Math. Soc., 105 (1962), 415-435. Zbl0111.11801MR30 #4164
  3. [3] R.I. JEWETT, A variation on the Stone-Weierstrass theorem, Proc. Amer. Math. Soc., 14 (1963), 690-693. Zbl0122.35004MR27 #2854
  4. [4] G. KLEINSTUCK, Der beschränkte Fall des gewichteten Approximationsproblems für vektorwertige Funktionen, Manuscripta Math., 17 (1975), 123-149. Zbl0343.41033MR53 #11364
  5. [5] L. NACHBIN, On the priority of algebras of continuous functions in weighted approximation, to appear in Symposia Mathematica. Zbl0338.41034
  6. [6] L. NACHBIN, Elements of Approximation Theory, D. van Nostran Co., Inc., 1967. Reprinted by R. Krieger Co., Inc., 1976. Zbl0173.41403
  7. [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. [8] J.B. PROLLA, Bishop's generalized Stone-Weierstrass theorem for weighted spaces, Math. Ann., 191 (1971), 283-289. Zbl0202.12603MR44 #7200
  9. [9] W. RUDIN, Real and complex analysis, McGraw-Hill, New York, 1966. Zbl0142.01701MR35 #1420
  10. [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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