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}, publisher = {Imprimerie Louis-Jean}, address = {Gap}, volume = {28}, number = {1}, year = {1978}, 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 TI - The general complex case of the Bernstein-Nachbin approximation problem JO - Annales de l'Institut Fourier PY - 1978 DA - 1978/// SP - 193 EP - 206 VL - 28 IS - 1 PB - Imprimerie Louis-Jean PP - Gap UR - https://aif.centre-mersenne.org/articles/10.5802/aif.685/ UR - https://zbmath.org/?q=an%3A0365.41007 UR - https://www.ams.org/mathscinet-getitem?mr=81g:46069 UR - https://doi.org/10.5802/aif.685 DO - 10.5802/aif.685 LA - en ID - AIF_1978__28_1_193_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.685/
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