H. Woracek:

"De Branges spaces and growth aspects";

in: "ASC Report 03/2014", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 30.

The subject of this survey is to review the basics of Louis de Branges´ theory of Hilbert spaces of entire functions, and to present results bringing together the notions of de Branges spaces on the one hand and growth functions (proximate orders) on the other.

After a few introductory words, the paper starts off with a short companion on de Branges theory (§2) where much of the terminology and cornerstones of the theory are presented. Then growth functions are -very briefly- introduced (§3). The following two section of the survey are devoted to growth properties. First (§4), some

general theorems, where the growth of elements of a de Branges space is discussed in relation with generating Hermite-Biehler functions and associated canonical systems, and results on growth of subspaces of a given space are presented. Second (§5), some

more concrete examples which appear "in nature", and where growth of different rates is exhibited. It should be said explicitly that this survey is of course far from being exhaustive.

For example, since the main purpose is to study growth properties of spaces of entire functions, all what relates to spectral measures (inclusion in L2-spaces, etc.) is omitted from the presentation.

http://www.asc.tuwien.ac.at/preprint/2014/asc03x2014.pdf

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