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