de Branges Spaces and Growth Aspects

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 one hand and growth functions (proximate orders) on the other hand. After a few introductory words, the paper starts off with a short companion on de Branges theory (section “A Short Companion on Hilbert Spaces of Entire Functions”) where much of the terminology and cornerstones of the theory are presented. Then growth functions are very briefly introduced (section “Growth Functions”). The following two sections of the survey are devoted to growth properties. First (section “General Theorems Relating de Branges Spaces and Growth”), 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 (section “Some Examples”), 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 L 2-spaces, etc.) is omitted from the presentation.