Frames in the Bargmann Hilbert space of entire functions. (English) Zbl 0632.30049

Daubechies, Ingrid; Grossman, A.

Commun. Pure Appl. Math. 41, No. 2, 151-164 (1988).

We look at the decomposition of arbitrary f in \(L^ 2({\mathbb{R}})\) in terms of the family of functions \(\phi _{mn}(x)=\pi ^{-1/4} \exp \{-mnab+i \max -(x-nb)^ 2\}\), with \(a,b>0\). We derive bounds and explicit formulas for the minimal expansion coefficients in the case where \(ab=(2\pi /N)\), N integer \(\geq 2\). Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form \[ m\| f\| ^ 2\leq \sum _{m,n\in Z}| f(z_ mn)| ^ 2 \exp ^{\{| z_{mn}| ^ 2\}}\leq M\| f\| ^ 2, \] where \(z_{mn}=ma+inb\), m and \(M>0\), and \(\| \|\) is the norm in F, \[ \| f\| ^ 2=(2\pi)^{-1}\iint _{R^ 2}dx dy| f(x+iy)| ^ 2 \exp ^{\{-(x^ 2+y^ 2)\}}. \] We conjecture that these inequalities remain true for all a,b such that \(ab<x2\pi\).

MSC:

30H05	Spaces of bounded analytic functions of one complex variable
46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Keywords:

Hilbert space

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