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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).

MSC:
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)
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