A Gabor system generated by a window function φ and a rectangular lattice aZ× Z/b is given by G(φ, aZ× Z/b) := {e−2πint/bφ(t−ma) : (m,n) ∈ Z× Z}. One of fundamental problems in Gabor analysis is to identify window functions φ and time-frequency shift lattices aZ × Z/b such that the corresponding Gabor system G(φ, aZ × Z/b) is a Gabor frame for L(R), the space of all square-integrable functions ...

In this paper we present a constructive proof that the set of Gabor frames is pathconnected in the L2(Rn)-norm. In particular, this result holds for the set of Gabor Parseval frames as well as for the set of Gabor orthonormal bases. In order to prove this result, we introduce a construction which shows exactly how to modify a Gabor frame or Parseval frame to obtain a new one with the same prope...

Title of dissertation: GABOR FRAMES FOR QUASICRYSTALS AND K-THEORY Michael Kreisel, Doctor of Philosophy, 2015 Dissertation directed by: Professor Jonathan Rosenberg Department of Mathematics We study the connection between Gabor frames for quasicrystals, the topology of the hull ΩΛ of a quasicrystal Λ and the K-theory of an associated twisted groupoid algebra. In particular, we construct a fin...

Let A ⊂ L2(R) be at most countable, and p, q ∈ N. We characterize various frame-properties for Gabor systems of the form G(1, p/q,A)= {e2 g(x − np/q) : m, n ∈ Z, g ∈ A} in terms of the corresponding frame properties for the row vectors in the Zibulski–Zeevi matrix. This extends work by [Ron and Shen, Weyl–Heisenberg systems and Riesz bases in L2(R d). Duke Math. J. 89 (1997) 237–282], who consi...

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for fr...

This talk addresses vector-valued subspace Gabor frames with rational time-frequency product. By introduction of a suitable Zak transform matrix, we characterize vector-valued subspace Gabor frames, Riesz bases and orthonorrmal bases, and the uniqueness of Gabor duals of type I and type II. Using the uniqueness results, we extend the classical Balian-Low theorem to vector-valued subspace Gabor ...

We study singly-generated wavelet systems on R that are naturally associated with rank-one wavelet systems on the Heisenberg group N . We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N , we give an explicit construction for Parseval frame wavelets that are associated with I . We say that g ∈ L(I×R) is Ga...