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Frames in the Bargmann space of entire functions. (English) Zbl 0770.30025
Entire and subharmonic functions, Adv. Sov. Math. 11, 167-180 (1992).
[For the entire collection see Zbl 0752.00059.]
Let \(B\) be the Hilbert space of entire functions with the scalar product \[ \langle f,g\rangle={1\over 2\pi}\iint_ \mathbb{C} f(z)\overline{g(z)} e^{-| z|^ 2} dm_ z. \] The author studies an opportunity of representation of functions from \(B\) by means of exponential series with exponents from \({\mathcal E}(Z)=\{e^{z_ \nu z/2}: z_ \nu\in Z\}\). Results are given in terms of an asymptotic behaviour of entire functions of the second order having a zero set \(Z\). Further results were proved recently by Yu. Lyubarskij and K. Seip, Ark. Mat. (to appear).

30D20 Entire functions of one complex variable, general theory
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)