TY - JOUR

T1 - De Branges' theorem on approximation problems of Bernstein type

AU - Baranov, A.

AU - Woracek, H.

PY - 2013

Y1 - 2013

N2 - 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. 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.

AB - 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. 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.

KW - weighted sup-norm approximation

KW - Bernstein type problem

KW - de Branges' theorem

U2 - 10.1017/S1474748013000030

DO - 10.1017/S1474748013000030

M3 - Article

VL - 12

SP - 879

EP - 899

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 4

ER -