On Hilbert spaces of sequences formed by values of functions from the Bargmann--Fock space

V. V. Napalkov ; A. A. Nuyatov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 104-114

Résumé

We study Hilbert spaces of sequences formed by values of functions from the Bargmann–Fock space $F$, which consists of entire functions whose square modulus is summable on the plane ${\mathbb C}$ with measure $d\sigma(z):= (1/\pi)e^{-|z|^2}\, dv(z)$, where $dv(z)$ is an area element: \begin{equation*} \|f\|^2_F=\int_{\mathbb C}|f(z)|^2\, d\sigma(z)\infty \quad \forall f\in F. \end{equation*} The space $\overline F$ consists of complex conjugates of functions from $F$, and $\|\overline f\|_{\overline F}=\|f\|_{F}\,\forall f\in F$. We consider classes of countable sets $\Omega_0\subset {\mathbb C}$ of the form $$ \Omega_0\stackrel{\mathrm{def}}{=}\{z\in{\mathbb C}\colon \ z=a\cdot n+ib\cdot m,\ a\cdot b=\pi\ \forall n,m\in {\mathbb Z}\}, $$ where $a$ and $b$ are some fixed (depending only on the set $\Omega_0$) nonzero real numbers. The sets $\Omega_0$ are called von Neumann lattices. For a real number $k>1$, we form the set $\Omega_0^k\stackrel{\mathrm{def}}{=}k\cdot\Omega_0$. We establish that the space of sequences of complex numbers $V_k$ formed by the traces of functions from some subspace $F^k$ of the space $F$ on the set $\Omega_0^k$ is equivalent to the space of sequences of complex numbers $U_k$ formed by the traces of functions from a subspace $\overline F^k$ of the space $\overline F$ on the set $\Omega_0^k$. The spaces $\overline F^k$ and $\overline F$ consist of complex conjugates of the functions from the spaces $F^k$ and $F$, respectively. Moreover, the norms in the spaces $V_k$ and $U_k$ are induced by the norms of the spaces $F^k$ and $\overline F^k$. To derive the main results of the paper, we use the result of K. Seip on discrete sampling sets of the Bargmann–Fock space. The results of the authors related to the questions of the coincidence or equivalence of Hilbert spaces with a reproducing kernel are applied. Here the notion of consistency of two complete systems of functions, introduced earlier by the authors, plays an important role. The paper presents counterexamples. We construct nonequivalent Hilbert spaces of complex numbers $V$ and $U$ that are the traces on some discrete subset of the complex plane of functions from $F$ that are not equivalent.

Détail

Keywords: decomposition systems similar to orthogonal ones, Hilbert space with reproducing kernel, problem of describing a dual space, Bargmann–Fock space, exponential frame, von Neumann lattice.