Wavelet representations and Fock space on positive matrices

On Parametrization of Compact Wavelet Matrices

It is given an efficient complete parametrization of wavelet matrices of rank $m$, genus $g+1$, and degree $g$, which are naturally identified with corresponding polynomial paraunitary matrix-functions. The parametrization depends on Wiener-Hopf factorization of unitary matrix-functions with constant determinant given in the unit circle. This method allows us to construct in real time the coefficients of wavelet matrices from the above class.

Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frames Atoms for L^2(R^d)

In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible …

Direct limits, multiresolution analyses, and wavelets

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on $L^2(\R^n)$, the wavelets on frac…

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $α\in \mathbb{R}$. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation no…

各向异性Balian-Low现象：小波框架的几何障碍

An Anisotropic Balian-Low Phenomenon: Geometric Obstructions to Wavelet Frames

本文研究扩张膨胀矩阵相关的各向异性Hardy空间中小波框架的解析稳定性，通过建立重建误差的确定性算子范数下界，揭示各向异性Balian-Low现象，其障碍源于几何不相容指数。

关于与不同伸缩群相关的小波共轨道空间

On wavelet coorbit spaces associated to different dilation groups

本文基于粗几何方法比较不同伸缩群定义的小波共轨道空间，为不可约和可约拟正则表示建立统一框架，揭示可约表示对纳入各向异性 Besov 空间等例子的必要性，并提供子群和共轨道空间等价性的刻画。