Wavelets on Fractals

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…

Radial multiresolution in dimension three

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in $\R^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in $\R^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on $\R$. Furthermore, decomposition and reconstruction algorithms are formulated.

$p$-Adic multiresolution analysis and wavelet frames

We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame.

On Parseval wavelet frames via Multiresolution Analyses

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Unitary Extension Principle and Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions.

A Family of Wavelets and a new Orthogonal Multiresolution Analysis Based on the Nyquist Criterion

A generalisation of the Shannon complex wavelet is introduced, which is related to raised cosine filters. This approach is used to derive a new family of orthogonal complex wavelets based on the Nyquist criterion for Intersymbolic Interference (ISI) elimination. An orthogonal Multiresolution Analysis (MRA) is presented, showing that the roll-off parameter should be kept below 1/3. The pass-band behaviour of the Wavelet Fourier spectrum is examined. The left and right roll-off regions are asymmetric; nevertheless the Q-constant analysis philosophy is maintained. Finally, a generalisation of the…

Vilenkin群上小波集的特征刻画及其在MRA小波构造中的应用

A characterization of wavelet sets on Vilenkin groups with its application to construction of MRA wavelets

本文在Vilenkin群上建立小波集的特征刻画，为小波构造提供新方法，并阐明多分辨分析与小波集确定的小波之间的关系，正面回答了Mahapatra和Singh提出的一个问题。