On interpolation and sampling in Hilbert spaces of analytic functions.

Abstract

In this paper we give new proofs of some theorems due to Seip Seip Wallst en and Lyubarskii Seip on sequences of interpolation and sampling for spaces of analytic functions that are square integrable with respect to certain weights The results are also given in a somewhat more general setting Introduction In a series of recent papers Seip S Seip Wallst en S W and Lyuabarskii Seip L S have studied sets of interpolation and sampling for various spaces of analytic functions of one variable Part of these results concern Hilbert spaces of functions that are square integrable against certain weights and another closely related part deals with similar spaces with uniform norms The methods used in the cited papers are based on classical type but intricate constructions of one variable nature that to some extent go back to Beurling B In O Ohsawa has suggested the use of L techniques for
to prove results of the above type In particular Ohsawa gives a proof of the su ciency part of the theorem of Seip Wallst en concerning interpolation in the space of entire functions in C satisfying Z jf j e jzj As in the approach initiated by Bombieri H ormander and Skoda see H the main di culty in such a proof is the construction of a pluri subharmonic function with prescribed singularities at the points where one wishes to interpolate For this Ohsawa uses part of the constructions of Seip Wallst en ultimately going back to Beurling and he poses as a problem to give a more elementary proof One purpose of this note is to show how that can be done As it turns out the method we use also works for more general weights and therefore also implies the su ciency part of the theorem by Lyubarskii Seip Furthermore we shall show how the positive direction of the sampling theorem can be obtained in a similar manner and we shall also permit somewhat more general growth conditions than Lyubarskii Seip see Theorem In O Ohsawa also gives a new proof of the theorem of Seip about interpolation in Bergman spaces of the disk This part of Ohsawa s work contains two essential ingredients The rst one is as in the case of entire space the construction of a subharmonic function First author supported by the NFR Second author was partially supported by the DGICYT grant PB C The research nec essary to conduct this work has been partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya