Frames in the bargmann space of entire functions

We look at the decomposition of arbitrary f in L 2 ( R ) in terms of the family of functions φ mn ( x ) = π −1/4 exp{ − 1/2 imnab + i max − 1/2( x − nb ) 2 }, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab = 2π/ N, N an integer ≧ 2. Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the formWe conjecture that these inequalities remain true for all a, b such that ab < 2π.