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De Branges’ theorem on approximation problems of Bernstein type

  • Anton Baranov (a1) and Harald Woracek (a2)

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted ${C}_{0} $ -space on the real line. A theorem of de Branges characterizes non-density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup-norm approximation by entire functions of exponential type at most $\tau $ and bounded on the real axis ( $\tau \gt 0$ fixed).

We consider approximation in weighted ${C}_{0} $ -spaces by functions belonging to a prescribed subspace of entire functions which is solely assumed to be invariant under division of zeros and passing from $F(z)$ to $ \overline{F( \overline{z} )} $ , and establish the precise analogue of de Branges’ theorem. For the proof we follow the lines of de Branges’ original proof, and employ some results of Pitt.

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1.Akhiezer, N. I., On weighted approximations of continuous functions by polynomials on the entire number axis. (Russian), Uspekhi Mat. Nauk (N.S.) 11 (4(70)) (1956), 343.
2.Akhiezer, N. I., The classical moment problem and some related topics in analysis (transl. from Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1961. English translation: Oliver & Boyd Edinburgh 1965.
3.Bakan, A. G., Representation of measures with polynomial denseness in , , and its applications to determinate moment problems, Proc. Amer. Math. Soc. 136 (10) (2008), 35793589.
4.Borichev, A. and Sodin, M., Weighted exponential approximation and non-classical orthogonal spectral measures, Adv. Math. 226 (3) (2011), 25032545. Branges, L., The Stone–Weierstrass theorem, Proc. Amer. Math. Soc. 10 (1959), 822824. Branges, L., The Bernstein problem, Proc. Amer. Math. Soc. 10 (1959), 825832.
7.Koosis, P., The logarithmic integral. I.Cambridge Studies in Advanced Mathematics, Volume 12 (Cambridge University Press, Cambridge, 1988).
8.Lubinsky, D. S., A survey of weighted polynomial approximation with exponential weights, Surv. Approx. Theory 3 (2007), 1105.
9.Mergelyan, S. N., Weighted approximations by polynomials (transl. from Russian), Uspekhi Mat. Nauk (N.S.) 11 (5(71)) (1956), 107152 English translation: 1958 American Mathematical Society Translations, Ser. 2, Volume 10 pp. 59–106.
10.Nachbin, L., Weighted approximation for algebras and modules of continuous functions: Real and self-adjoint complex cases, Ann. of Math. (2) 81 (1965), 289302.
11.Pitt, L. D., A general approach to approximation problems of the Bernstein type, Adv. Math. 49 (3) (1983), 264299.
12.Pollard, H., Solution of Bernstein’s approximation problem, Proc. Amer. Math. Soc. 4 (1953), 869875.
13.Poltoratski, A., Bernstein’s problem on weighted polynomial approximation (arXiv:1110.2540v2) 28 Oct. 2011.
14.Rudin, W., Real and complex analysis, Third edition. (McGraw-Hill Book Co, New York, 1987).
15.Sodin, M. and Yuditskii, P., Another approach to de Branges’ theorem on weighted polynomial approximation, Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), 221–227, Israel Math. Conf. Proc., 11, Bar-Ilan Univ., Ramat Gan, 1997.
16.Summers, W. H., A representation theorem for biequicontinuous completed tensor products of weighted spaces, Trans. Amer. Math. Soc. 146 (1969), 121131.
17.Summers, W. H., Dual spaces of weighted spaces, Trans. Amer. Math. Soc. 151 (1970), 323333.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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