We study invertibility and compactness of positive Toeplitz operators associated to a continuous Parseval frame on Hilbert space. As applications, we characterize affine Weyl-Heisenberg localization as well give uncertainty principles for the transforms.

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 construct Shannon-like Parseval frame wavelets on a class of non commutative two-step nilpotent Lie groups. Our work was inspired by a construction given by Azita Mayeli on the Heisenberg group. The tools used here are representation theoretic. However, a great deal of Gabor theory is used for the construction of the wavelets. The construction obtained here is very explicit, and we are even ...

A SIC consists of N equiangular unit vectors in an N dimensional Hilbert space. The frame potential is a function of N unit vectors. It has a unique global minimum if the vectors form a SIC, and this property has been made use of in numerical searches for SICs. When the vectors form an orbit of the Heisenberg group the frame potential becomes a function of a single fiducial vector. We analytica...

We show that the conjectured generalization of the BourgainTzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linear independent sequences. We further show that the two conjectures are impl...

In this paper, we characterize the space of almost periodic (AP ) functions in one variable using either a Weyl-Heisenberg (WH) system or an affine system. Our observation is that the sought-for characterization of the AP space is valid if and only if the given WH (respectively, affine) system is an L2(IR)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our chara...

for all f ∈ H . The constant A (respectively, B) is a lower (resp. upper) frame bound for the frame. One of the most important frames for applications, especially signal processing, are the Weyl-Heisenberg frames. For g ∈ L(R) we define the translation parameter a > 0 and the modulation parameter b > 0 by: Embg(t) = e , Tnag(t) = g(t− na). For g ∈ L(R) and a, b > 0, we say for short that (g, a,...

AWeyl-Heisenberg frame {EmbTnag}m,n∈Z = {eg(·−na)}m,n∈Z for L2(R) allows every function f ∈ L2(R) to be written as an infinite linear combination of translated and modulated versions of the fixed function g ∈ L2(R). In the present paper we find sufficient conditions for {EmbTnag}m,n∈Z to be a frame for span{EmbTnag}m,n∈Z , which, in general, might just be a subspace of L2(R) . Even our conditio...

In [NeuroQuantology 6, 46 (2008): arXiv:0706.4180], a certain natural phenomenon of self-observing consciousness was shown to be incompatible with the standard axioms of quantum theory. In order to consider a possibility of removing this inconsistency, which arises due to the separation between an observing party and a state vector being observed stated in the quantum theory axioms, we provide ...

Using the Monte Carlo method in the frame of the Heisenberg model we performed simulation of magnetic properties of Re100-xTbx (x=5–91 at. %) amorphous alloys and of pure amorphous Tb. Exchange interaction between the nearest neighbours and random uniaxial anisotropy were taken into account in the Hamiltonian. We calculated temperature dependence of magnetization, Edwards–Anderson order paramet...