Heisenberg Frame Sets

Proceedings of the American Mathematical Society

A Weyl-Heisenberg frame {E mb Tnag} m,n∈Z = {e 2πimb(•) g(• − na)} m,n∈Z for L 2 (R) allows every function f ∈ L 2 (R) to be written as an infinite linear combination of translated and modulated versions of the fixed function g ∈ L 2 (R). In the present paper we find sufficient conditions for {E mb Tnag} m,n∈Z to be a frame for span{E mb Tnag} m,n∈Z , which, in general, might just be a subspace of L 2 (R). Even our condition for {E mb Tnag} m,n∈Z to be a frame for L 2 (R) is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for L 2 (R), showing for instance that the condition G(x) := n∈Z |g(x − na)| 2 > A > 0 is not necessary for {E mb Tnag} m,n∈Z to be a frame for span{E mb Tnag} m,n∈Z. Our work is inspired by a recent paper by Benedetto and Li, where the relationship between the zero-set of the function G and frame properties of the set of functions {g(• − n)} n∈Z is analyzed.

Sampling, Wavelets, and Tomography, 2004

We give an introduction to irregular Weyl-Heisenberg frames showing the latest developments and open problems. We provide several new results for semiirregular WH-frames as well as giving new and more accessable proofs for several results from the literature.

Applied and Computational Harmonic Analysis, 2004

From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (g mα,nβ ) m,n∈Z for L 2 (R) has a unique WH-dual if and only if αβ = 1. However, the same argument does not apply to the subspace WH-frame case and it is not clear how to use standard methods of Fourier analysis to deal with this situation. In this paper, we apply operator algebra theory to obtain a very simple necessary and sufficient condition for a given frame (induced by a projective unitary representation of a discrete group) to admit a unique dual (induced by the same system). As a special case, we obtain a characterization for the subspace WH-frames that have unique WH-duals (within the subspace). Using this characterization and the Zak transform, we are able to prove that if (g mα,nβ ) m,n∈Z is a WH-frame for a subspace M of L 2 (R), then, (i) (g mα,nβ ) m,n∈Z has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (g mα,nβ ) m,n∈Z has a unique WH-dual in M if and only if (g mα,nβ ) m,n∈Z is a Riesz sequence; (iii) if αβ < 1, then the WH-dual for (g mα,nβ ) m,n∈Z in M is not unique.  2004 Elsevier Inc. All rights reserved.

2010

Let {\phi} be an arbitrary generalized Gaussian (squeezed coherent state), {\Lambda}_{{\alpha}{\beta}}=({\alpha}_1 Z \times\cdot\cdot\cdot\times \alpha_{n}\mathbb{Z)\times}(\beta_{1}\mathbb{Z}\times\cdot\cdot\cdot \times\beta_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on {\phi}) such that the multivariate Weyl--Heisenberg system G({\phi},M{\Lambda}_{{\alpha}{\beta}}) is a frame. In a forthcoming Note we will prove a converse to this result.

2010

In a previous Note we established a necessary and sufficient condition for a multivariate Weyl--Heisenberg system G({\phi},{\Lambda}) to be a frame when the window is a generalized Gaussian (squeezed coherent state) and {\Lambda} a rectangular lattice. In this Note we extend this result to lattices of the type ${\Lambda}=MZ^{2n}$ where M is a positive definite symmetric matrix.

Advances in Gabor Analysis, 2003

We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions (g n ) into a sequence (e n ) with the property that (E mb e n ) m,n∈Z is orthonormal in L 2 (R). Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions g ∈ L 2 (R) and ab = 1 so that the family (E mb T na g) is complete in L 2 (R). One consequence of this is that for functions g supported on a half-line [α, ∞) (in particular, for compactly supported g), (g, 1, 1) is complete if and only if sup 0≤t<a |g(t − n)| = 0 a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any g ∈ L 2 (R), A ≤ n |g(t − na)| 2 ≤ B is equivalent to (E m/a g) being a Riesz basic sequence.

Wavelet Applications in Signal and Image Processing VIII, 2000

We examine the question of which characteristic functions yield Weyl-Heisenberg frames for various values of the parameters. We also give numerous applications of frames of characteristic functions to the general case (g, a, b).

We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., finding a set of d 2 unit vectors (φ j) in C d with | φ j , φ k | = 1 √ d + 1 , j = k. Such 'equally spaced configurations' have appeared in various guises, e.g., as complex spherical 2-designs, equiangular tight frames, isometric embeddings 2 (d) → 4 (d 2), and most recently as SICPOVMs in quantum measurement theory. Analytic solutions are known only for d = 2, 3, 4, 8. Recently, numerical solutions which are the orbit of a discrete Heisenberg group H have been constructed for d ≤ 45. We call these Heisenberg frames. In this paper we study the normaliser of H, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for d = 5, 7, and make conjectures about the form of a solution when d is odd. Most notably, it appears that solutions for d odd are eigenvectors of some element in the normaliser which has (scalar) order 3.

Elementary theory of Fourier series tells us that the exponential functions e m (x) = e 2πımx , m ∈ Z, form an orthonormal basis for the space of square integrable functions of period one on the real line. This is equivalent to saying that {χ.e m : m ∈ Z} is an orthonormal basis for the space M 0 of square integrable functions on the real line that vanish outside the unit interval [0,1), where χ = χ [0,1) is the characteristic function of the interval. Thus, if T n are the translation operators defined by T n f (x) = f (x −n), then {e m .T n χ } is an orthonormal basis for M n , the subspace of L 2 (R) consisting of functions supported on [n, n + 1), n ∈ Z. Since L 2 (R) is the orthogonal direct sum ⊕ n∈Z M n , we get an orthonormal basis {e m .T n χ : n, m ∈ Z} for

Applied and Computational Harmonic Analysis, 2002

In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If g ∈ W (L ∞ , L 1 ), then for all continuous, compactly supported functions f we have: