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De Branges’ theorem on approximation problems of Bernstein type

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

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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References
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A. Borichev and M. Sodin , Weighted exponential approximation and non-classical orthogonal spectral measures, Adv. Math. 226 (3) (2011), 25032545.

L. de Branges , The Stone–Weierstrass theorem, Proc. Amer. Math. Soc. 10 (1959), 822824.

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L. Nachbin , Weighted approximation for algebras and modules of continuous functions: Real and self-adjoint complex cases, Ann. of Math. (2) 81 (1965), 289302.

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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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