151 research outputs found
Classification of Finite Group-Frames and Super-Frames
No full textAbstractGiven a finite group G, we examine the classification of all frame representations of G and the classification of all G-frames, i.e., frames induced by group representations of G. We show that the exact number of equivalence classes of G-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number L such that there exists an L-tuple of strongly disjoint G-frames.</jats:p
The existence of tight Gabor duals for Gabor frames and subspace Gabor frames
No full textAbstractLet K and L be two full-rank lattices in Rd. We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time–frequency lattice K×L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K×L is less than or equal to 12. (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume v(K×L)⩽12 or v(K×L)⩾2. Moreover, if K=αZd, L=βZd with αβ=1, then a subspace Gabor frame G(g,L,K) has a tight Gabor pseudo-dual only when G(g,L,K) itself is already tight
>
No full textGiven a window function which generates a Gabor (resp. wavelet) frame. We consider the best approximation by those window functions that generate normalized tight (or just tight) frames. Using a parameterizations of window functions by certain class of operators in the von Neumann algebras associated with shift operators in time and frequency over certain lattices, we are able to prove that for any window function of a Gabor frame, there exists a unique window function which generates a tight Gabor frame and is the best approximation (among all the tight Gabor frames) for the given window function. More generally, we show that this is true for any frame induced by a projective unitary representation for a group. However, this result is not valid for wavelet frames. We will provide a restricted approximation result for semi-orthogonal wavelet frames
Dilations And Completions For Gabor Systems
No full textLet Λ=K×L be a full rank time-frequency lattice in a, d ×a, d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L 2(a, d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)\u3e1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419-433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel-Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ) ( j=1N G(g j ,Λ)) for L 2(a, d ). We show that this is true whenever v(Λ) N. In particular, when v(Λ) 1, any Bessel-Gabor system is a subset of a tight Gabor frame G(g,Λ) G(h,Λ) for L 2(a, d ). Related results for affine systems are also discussed. © 2008 Birkhäuser Boston
A Note On The Density Theorem For Projective Unitary Representations
No full textIt is well known that a Gabor representation on L2(ℝd) admits a frame generator h ∈ L2(ℝd) if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices
Interpolation Operators Associated With Sub-Frame Sets
No full textInterpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the local commutant space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the 2π-congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same 2π-congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set
Frame Representations And Parseval Duals With Applications To Gabor Frames
No full textLet {xn} be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {xn} which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {xn} can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame {π (g) ξ : g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π (g) ξ : g ∈ G} can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame {π (g) ξ : g ∈ G} (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame G(g,L,K) (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of L×K is less than or equal to 1 2 . Copyright © 2008 American Mathematical Society
On isomorphisms and hyper-reflexivity of closed subspace lattices
No full textThere are some papers, such as [1], [2] and [3], in which some properties on
isomorphism of closed subspace lattices of Hilbert spaces were studied. In this short paper we will
point out that the hyper-reflexivity of closed subspace lattice is invariant under isomorphism of
ξ(H1) on ξ(H2). We also proved that if T is in L(H) such that 0∈¯π(T) and ℱ is a hyper-reflexive
subspace lattice, then ϕT(ℱ)∪{H} is hyper-reflexive where ϕT is a homomorphism induced by T
Tight frame approximation for multi-frames and super-frames
No full textAbstractWe consider a generator Φ=(φ1,…,φN) for either a multi-frame or a super-frame generated under the action of a projective unitary representation for a discrete countable group. Examples of such frames include Gabor multi-frames, Gabor super-frames and frames for shift-invariant subspaces. We show that there exists a unique normalized tight multi-frame (resp. super-frame) generator Ψ=(ψ1,…,ψN) such that ∑j=1N||φj−ψj||2⩽∑j=1N||φj−ψj||2 holds for all the normalized tight multi-frame (resp. super-frame) generators η=(η1,…,ηN). We also investigate the similar problems for dual frames and discuss a few applications to Gabor frames and some other frames
- …
