Crystallographic Multiwavelets in $L^2(R^d)$

TL;DR

This paper characterizes the scaling functions of crystal multiresolution analyses in $L^2(R^d)$, providing conditions for the existence of associated crystal wavelet bases based on symbol matrices.

Contribution

It introduces a characterization of crystal multiresolution analysis scaling functions and establishes necessary and sufficient conditions for crystal wavelet basis existence.

Findings

Provides a characterization of the scaling function in terms of vector-scaling functions.

Establishes necessary and sufficient conditions via symbol matrices for wavelet basis existence.

Connects multiresolution analysis in $L^2(R^d)$ with lattice-based structures.

Abstract

We characterize the scaling function of a crystal Multiresolution Analysis in terms of the vector-scaling function for a Multiresolution Analysis associated to a lattice. We give necessary and sufficient conditions in terms of the symbol matrix in order that an associated crystal wavelet basis exists.