Weyl - Heisenberg frames and Riesz bases in L 2 ( IR d )
نویسندگان
چکیده
We study Weyl-Heisenberg (=Gabor) expansions for either L2(IR ) or a subspace of it. These are expansions in terms of the spanning set X = (EM φ : k ∈ K, l ∈ L,φ ∈ Φ), where K and L are some discrete lattices in IR, Φ ⊂ L2(IR ) is finite, E is the translation operator, and M is the modulation operator. Such sets X are known as WH systems. The analysis of the “basis” properties of WH systems (e.g. being a frame or a Riesz basis) is our central topic, with the fiberization-decomposition techniques of shift-invariant systems, developed in a previous paper of us, being the main tool. Of particular interest is the notion of the adjoint of a WH set, and the duality principle which characterizes a WH (tight) frame in term of the stability (orthonormality) of its adjoint. The actions of passing to the adjoint and passing to the dual system commute, hence the dual WH frame can be computed via the dual basis of the adjoint. Estimates for the underlying frame/basis bounds are obtained by two different methods. The Gramian analysis applies to all WH systems, albeit provides estimates that might be quite crude. This approach is invoked to show how, under only mild conditions on X, a frame can be obtained by oversampling a Bessel set. Finally, finer estimates of the frame bounds, based on the Zak transform, are obtained for a large collection of WH systems. AMS (MOS) Subject Classifications: Primary 42C15, Secondary 42C30
منابع مشابه
Weyl-heisenberg Frames and Riesz Bases in L 2 (ir D ) Weyl-heisenberg Frames and Riesz Bases in L 2 (ir D )
We study Weyl-Heisenberg (=Gabor) expansions for either L 2 (IR d) or a subspace of it. These are expansions in terms of the spanning set where K and L are some discrete lattices in IR d , L 2 (IR d) is nite, E is the translation operator, and M is the modulation operator. Such sets X are known as WH systems. The analysis of the \basis" properties of WH systems (e.g. being a frame or a Riesz ba...
متن کاملWeyl - Heisenberg frames and Riesz bases in L 2 ( IRd ) Amos
We study Weyl-Heisenberg (=Gabor) expansions for either L 2 (IR d) or a subspace of it. These are expansions in terms of the spanning set where K and L are some discrete lattices in IR d , L 2 (IR d) is nite, E is the translation operator, and M is the modulation operator. Such sets X are known as WH systems. The analysis of the \basis" properties of WH systems (e.g. being a frame or a Riesz ba...
متن کاملA characterization of L-dual frames and L-dual Riesz bases
This paper is an investigation of $L$-dual frames with respect to a function-valued inner product, the so called $L$-bracket product on $L^{2}(G)$, where G is a locally compact abelian group with a uniform lattice $L$. We show that several well known theorems for dual frames and dual Riesz bases in a Hilbert space remain valid for $L$-dual frames and $L$-dual Riesz bases in $L^{2}(G)$.
متن کامل