# Solutions to dilation equations

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0111.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**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.