De Branges’ theorem on approximation problems of Bernstein type

Publisher： Cambridge University Press

E-ISSN： 1475-3030|12|4|879-899

ISSN： 1474-7480

Source： Journal of the Institute of Mathematics of Jussieu, Vol.12, Iss.4, 2013-10, pp. : 879-899

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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).

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