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...

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. A fundamental set of equalities are shown ...

An independent set S of a connected graph G is called a frame if G − S is connected. If |S| = k, then S is called a k-frame. We prove the following theorem. Let k ≥ 2 be an integer, G be a connected graph with V (G) = {v1, v2, . . . , vn}, and degG(u) denote the degree of a vertex u. Suppose that for every 3-frame S = {va, vb, vc} such that 1 ≤ a < b < c ≤ n, degG(va) ≤ a, degG(vb) ≤ b − 1 and ...

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

Hydrofluorocarbon-based (HFC) fire extinguishing agents generate hydrogen fluoride (HF) gas [HF(g)] during thermal decomposition. Potential HF(g) exposure represents a health risk to any personnel not wearing proper Personal Protective Equipment (PPE) scenarios where HFC extinguishers are deployed. is harmful humans at vapor concentrations greater than 3 parts per million (ppm), with an Immedia...

In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this note it is shown that it actually depends critically on the timefrequency lattice used. Indeed, for any irrational > 0 we can construct a smooth g 2 L2(R) such that for any two rationals a > 0 and b > 0 the collection (gna;mb)n;m2Zof time-frequency translates of g has a finite...

Nomenclature β,γ(∈ S) = The gimbal and wheel angles. (rad) Rβ(∈ SO(3)) = Transformation from gimbal frame G to the spacecraft body frame B. Is = Spacecraft inertia without the CMG gimbal and wheel inertia. (kg.m) Ig, Ir = Gimbal frame inertia, wheel inertia about own centre of mass represented in gimbal frame. (kg.m) (Igr)β = Combined inertia of gimbal frame and wheel in the spacecraft frame. R...

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...