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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)
##### Keywords:
Bargmann space; Hilbert space; exponential series