EL-FALLAH, O; FRICAIN, E; KELLAY, Karim; MASHREGHI, J; RANSFORD, T.

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EL-FALLAH, O
Département de Mathématiques

FRICAIN, E
Laboratoire Paul Painlevé - UMR 8524 [LPP]

KELLAY, Karim
Institut de Mathématiques de Bordeaux [IMB]

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Constructive Approximation. 2016-09-21, vol. 44, n° 2, p. 269-281

Springer Verlag

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In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f_r(z):=f(rz)~(r < 1)$.We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim_{r\to1^-}\|f_r\|_{\cH(b)}=\infty$.It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this fact.< Leer menos

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Constructive approximation in de Branges-Rovnyak spaces

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