Solutions to dilation equations

We develop a general technique which is applied to classify all L2 solutions of the Haar dilation equation. This leads to a result characterising operators which commute with shifts and dilations.

We also present a new tool for studying compactly-supported solutions and derive several interesting results concerning refinable characteristic functions, 2- and 3-refinable functions, smoothness and boundedness.

Finally we present a few shorter results: demonstrating how to find polynomial solutions to dilation equations, showing how to combine dilation equations and their solutions, and determining when a self-affine tile can be a parallelepiped.