MULTIWAVELETS IN Rn WITH AN ARBITRARY DILATION MATRIX

We present an outline of how the ideas of self-similarity can be applied to wavelet theory, especially in connection to wavelets associated with a multiresolution analysis of R n allowing arbitrary dilation matrices and no restrictions on the number of scaling functions. 1. Introduction Wavelet bases have proved highly useful in many areas of mathematics, science, and engineering. One of the most successful approaches for the construction of such a basis begins with a special functional equation, the reenement equation. The solution to this reenement equation, called the scaling function, then determines a multiresolution analysis, which in turn determines the wavelet and the wavelet basis. In order to construct wavelet bases with prescribed properties, we must characterize those particular reenement equations which yield scaling functions that possess some speciic desirable property. Much literature has been written on this topic for the classical one-dimensional, single-function, two-scale reenement equation, but when we move from the one-dimensional to the higher-dimensional setting or from the single wavelet to the multiwavelet setting it becomes increasingly diicult to nd and apply such characterizations. Our goal in this paper is to outline some recent developments in the construction of higher-dimensional wavelet bases that exploit the fact that the reenement equation is a statement that the scaling function satisses a certain kind of self-similarity. In the classical one-dimensional case with dilation factor two, there are a variety of tools in addition to self-similarity which can be used to analyze the reenement equation. However, many of these tools become diicult or impossible to apply in the multidimensional setting with a general dilation matrix, whereas self-similarity becomes an even more natural and important tool in this setting. By viewing scaling functions as particular cases of \generalized self-similar functions," we showed in CHM99] that the tools of functional analysis can be applied to analyze reenement equations in the general higher-dimensional and multi-function setting. We derived conditions for the existence of continuous or L p solutions to the reenement equation in this general setting, and showed how these conditions can be combined with the analysis of the accuracy of scaling functions from CHM98], CHM97] to construct new examples of