Frames in the bargmann space of entire functions

Abstract

We look at the decomposition of arbitrary f in L²(R) in terms of the family of functions φ_mn(x) = π^−1/4exp{ − 1/2imnab + 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 form We conjecture that these inequalities remain true for all a, b such that ab < 2π.