On the Hermite expansions of functions from the Hardy class

Abstract

Considering functions f on R n for which both f and f are bounded by the Gaussian e − 1 2 a|x| 2 , 0 < a < 1 we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for O(n)−finite functions thus extending the one dimensional result of Vemuri [11].

On the structure of analytic vectors for the Schrödinger representation

“…It is conjectured that under the hypothesis of Theorem 4.7 (which corresponds to a = b = tanh 2t < 1 in Hardy's theorem) the Fourier-Hermite coefficients of ψ decay like e −(2|α|+n)t/2 . In [3] we have proved this result under the extra assumption that the spherical harmonic expansion of f is finite. In the general case the conjecture is still open.…”

“…In [4] we have proved this result under the extra assumption that the spherical harmonic expansion of f is finite. In the general case the conjecture is still open.…”

“…Proof We deduce the result from the one dimensional case which is due to Vemuri [20] (see also [4]). Assuming the one dimensional result consider where α = (μ, k) ∈ N n−1 × N, (x, y) ∈ R n−1 × R and…”

Analyticity of the Schrödinger propagator on the Heisenberg group

“…α with eigenvalues (2k + α + 1)|λ|. The images of L 2 under the semigroups for Bessel, Hermite, Laguerre operators have been characterized in [2,7,12,27]. We will be using some results stated and proved there.…”

Variations on a theorem of Beurling

“…In view of this it is natural to ask for a characterisation of all functions f satisfying K a (f ) < ∞ for a fixed 0 < a < 1. An analogue of this problem in the context of Hardy's theorem has been studied by Demange [2], Vemuri [10] and the authors [5]. Recall the statement of Hardy's theorem [3]: If |f (x)| ≤ Ce −a|x| 2 , |f (y)| ≤ Ce −b|y| 2 then (i) f = 0 when ab > 1 4 , (ii) f (x) = Ce −a|x| 2 when ab = 1 4 and (iii) when ab < 1 4 there are infinitely many linearly independent functions (e.g.…”