Series: Mathematics Colloquium

Date: Thursday, April 17, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Winnie Li

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Jeffrey Lagarias, AT&T Research

Title: Wavelets, Tilings, and Number Theory

Abstract:

This talk considers orthonormal wavelet bases of the Hilbert space of
square-summable functions on n-dimensional Euclidean space.  These are
orthonormal bases formed by translates and dilations of a single
function; the Haar basis is the prototypical example.  Such wavelets
are specified by a scaling function, which is a solution of a
functional difference equation, called a dilation equation.  This
equation involves a dilation map which takes x to Mx, where M is an
integer n by n matrix which is expanding, meaning all its eigenvalues
are of length exceeding one. Ingrid Daubechies showed there exist
orthonormal bases of compactly supported wavelets of arbitrary
smoothness for dilations taking x to 2x on the line.  Do such wavelets
exist for all dilation matrices M? We consider the case of Haar-type
wavelets. Their existence is related to radix expansions to base M
having nice tiling properties. These lead to problems in number
theory, some solved and some unsolved.