Frames in the Bargmann space of entire functions. (English) Zbl 0770.30025

Lyubarskij, Yu. I

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).

Reviewer: M.Sodin (Khar’kov)

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

Citations:

Zbl 0752.00059