Radial multiresolution in dimension three

Hermite 矩阵谱上的多分辨率分析

Multiresolution analysis on spectra of hermitian matrices

本文在复 Hermite 矩阵空间上构造适应酉群共轭不变性的多分辨率分析，利用超群平移刻画并证明径向标准正交小波基的存在性。

On compactly supported discrete radial wavelets in $L^2(\mathbb{R}^2)$ and application in Tomography

Radially symmetric wavelets possessing multiresolution framework are found to be useful in different fields like Pattern recognition, Computed Tomography (CT) etc. The compactly supported wavelets are known to be useful for localized operations in applications such as reconstruction, enhancement etc. In this work we introduce a novel way of designing compactly supported radial wavelets in $L^2(\mathbb{R}^2)$ from a 1D Daubechies wavelets and obtain a reconstruction formula possessing multiresolution framework. Further, we demonstrate the usefulness of our radial wavelets in Tomography.

Projective multiresolution analyses for dilations in higher dimensions

We continue the study of projective module wavelet frames corresponding to diagonal dilation matrices on $\mathbb R^n$ with integer entries, focusing on the construction of a projective multi-resolution analysis corresponding to dilations whose domains are finitely generated projective modules over continuous complex-valued functions on $n$-tori, $n\geq 3$. We are able to generalize some of these results to dilation matrices that are conjugates of integral diagonal dilation matrices by elements of $SL(n,\mathbb Z).$ We follow the method proposed by the author and M. Rieffel, and are able to co…

紧支撑插值张量积小波的多分辨率分析（长版）

Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)

本研究运用巴拿赫空间张量积理论，为两类n维函数空间构建插值张量积MRA及其对偶MRA，并将Besov空间范数等价结果推广到n维情形。

Wavelets on Fractals

We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on the line, there are also sharp contrasts. These are stated in our main result, a dichotomy theorem. The first section is the case of the middle-third Cantor set. This is followed by a review of the essentials on Hausdorff measure. The remaining sections of the paper cover multiresolutions in the general context of affine iterated function systems.

Representation theory and multilevel filters

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the unitary group $U_N$. We call $G$ the loop group because of the special case of the unit circle.