Narrow Results

A WH-packet is a system of vectors which is analogous to Aldroubi’s model for explicit expression of vectors (including frame vectors) in terms of a series associated with a given frame. In this paper, we study frame properties of WH-packet type system for matrix-valued wave packet frames in the function space $L^2({ℝ}^d,{ℂ}^{s\times r})$. A necessary and sufficient condition for WH-packets of matrix-valued wave packet frames in terms of a bounded below operator is given. We present sufficient conditions for both lower and upper frame conditions on scalars associated with WH-packet of matrix-valued wave packet frames. Finally, a Paley–Wiener type perturbation theorem for WH-packet of matrix-valued wave packet frames is given. Several examples are given to illustrate the results.

In this paper, we introduce some new $K$-frames related to the adjoint of a $K$-frame and sum of two $K$-frames in Hilbert spaces. We prove that adjoint of a normal $K$-frame is also a $K$-frame.

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator $\mathrm{\Theta }$ on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate $\mathrm{\Theta }$-Gabor frame. By extending work of G$\check{\mathrm{\text{a}}}$vruta, we present necessary and sufficient conditions for the existence of $\mathrm{\Theta }$-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and $\mathrm{\Theta }$-Gabor frames into $\mathrm{\Theta }$-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued $\mathrm{\Theta }$-Gabor frames.

We show that for every nonzero vector $\varphi$ in ${ℂ}^N$, the discrete wave packet system ${\{D_a{M}_l{T}_k\varphi \}}_{a\in U(N),l,k\in {𝕀}_N}$ constitutes a frame for the unitary space ${ℂ}^N$. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in ${ℂ}^{\mathbb{N}}$. The canonical dual of discrete wave packet frame is also discussed.

Let $G$ be a locally compact group. This work aims to study some important properties of the frame of translates for the space $L^2(G)$. Also, we showed that the frame of translates can be a frame for a proper subspace of $L^2(G)$, in the case that $G$ has a contractive automorphism. Furthermore, we characterize the frame properties for such systems and find an expression for the canonical dual frame.

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space $L^2({ℝ}^d,{ℂ}^{s\times r})$ from dual pairs of atomic wave packet frames in $L^2({ℝ}^d)$ is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of $L^2({ℝ}^d)$.

In recent years, the concepts of frame in $n$-Hilbert space and $K$-frames in $n$-Hilbert space have been introduced. The $K$-frames in the $n$-Hilbert space are more generalized than the ordinary frames. This paper deals with the problem of sum of $K$-frames in $n$-Hilbert space. First, the sufficient condition for the finite sum of $K$-frames and Bessel sequences in $n$-Hilbert space is given. Then, some results on the operator $K$ and pre-frame operator are given, and some new results on the sum of $K$-frame are proved. We then discuss several results for the perturbation and stability of the $K$-frames in $n$-Hilbert space about the sum of the $K$-frames.