De Branges' theorem on approximation problems of Bernstein type

TL;DR

This paper extends de Branges' theorem to approximation problems of Bernstein type, characterizing density in weighted spaces via entire functions of Krein class and invariance properties, using classical proof techniques.

Contribution

It generalizes de Branges' theorem to broader classes of entire functions with specific invariance properties in weighted approximation spaces.

Findings

Characterization of non-density via Krein class functions.

Extension of de Branges' theorem to invariant subspaces of entire functions.

Use of classical proof methods and Pitt's results to establish the analogue.

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 L. 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 ($\tau>0$ 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 $\bar{F(\bar z)}$, 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 L. Pitt.