Gabor (super)frames with Hermite functions

Gabor (super)frames with Hermite Functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions Hn. Let h = (H0, H1, . . . , Hn) be the vector of the first n+ 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R such that the Gabor system {e2πiλ2th(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a frame for L(R,C). As a corollary we obtain sufficient conditions for a single H...

Simultaneous estimates for vector-valued Gabor frames of Hermite functions

We derive frame estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides frame bound estimates for windows composed of Hermite functions. The proof is based on a recently established sampling theorem for the simply connected Heisenberg group, which is translated to a family of frame estimates via a direct integral decomposition.

A note on reassigned Gabor spectrograms of Hermite functions

An explicit form is given for the reassigned Gabor spectrogram of an Hermite function of arbitrary order. It is shown that the energy concentration sharply localizes outside the border of a clearance area limited by the “classical” circle where the Gabor spectrogram attains its maximum value, with a perfect localization that can only be achieved in the limit of infinite order. 1 Motivation 1.1 ...

Hermite functions

We define S to be the set of those φ ∈ C∞(R,C) such that pn(φ) < ∞ for all n ≥ 0. S is a complex vector space and each pn is a norm, and because each pn is a norm, a fortiori {pn : n ≥ 0} is a separating family of seminorms. With the topology induced by this family of seminorms, S is a Fréchet space. As well, D : S → S defined by (Dφ)(x) = φ′(x), x ∈ R 1http://individual.utoronto.ca/jordanbell/...

Schwartz functions, Hermite functions, and the Hermite operator