George C. Donovan

Construction of Orthogonal Wavelets Using Fractal Interpolation Functions

Fractal interpolation functions are used to construct a compactly supported continuous, orthogonal wavelet basis spanning

Fractal function, splines, intertwining multiresolution analysis, and wavelets

L^2 (\mathbb{R})

Squeezable Orthogonal Bases: Accuracy and Smoothness

. The wavelets share many of the properties normally associated with spline wavelets, in particular, they have linear phase.

C0 spline wavelets with arbitrary approximation order

The construction of smooth, orthogonal compactly supported wavelets is accomplished using fractal interpolation functions and splines. These give rise to multiwavelets. In the latter case piecewise polynomial wavelets are exhibited using an intertwining multiresolution analysis.

Construction of two-dimensional multiwavelets on a triangulation

We present a method for generating local orthogonal bases on arbitrary partitions of R from a given local orthogonal shift-invariant basis via what we call a squeeze map. We give necessary and sufficient conditions for a squeeze map to generate a nonuniform basis that preserves any smoothness and/or accuracy (polynomial reproduction) of the shift-invariant basis. When the shift-invariant basis has sufficient smoothness or accuracy, there is a unique squeeze map associated with a given partition that preserves this property and, in this case, the squeeze map may be calculated locally in terms of the ratios of adjacent intervals. If both the smoothness and accuracy are large enough, then the resulting nonuniform space contains the nonuniform spline space characterized by that smoothness and accuracy. Our examples include a multiresolution on nonuniform partitions such that each space has a local orthogonal basis consisting of continuous piecewise quadratic functions. We also construct a family of smooth, local, orthogonal, piecewise polynomial generators with arbitrary approximation order.

Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets

The theory of orthogonal polynomials is used to construct a family of orthogonal wavelet bases of L2(R) which are compactly supported, continuous, and piecewise polynomial and have arbitrary approximation order.

Compactly Supported, Piecewise Affine Scaling Functions on Triangulations

A family of continuous, compactly supported, bivariate multi-scaling functions have recently been constructed by Donovan, Geronimo, and Hardin using self-affine fractal surfaces.In this paper we describe a construction of associated multiwavelets that uses the symmetry properties of the multi-scaling functions. Illustrations of a particular set of scaling functions and wavelets are provided.