From: Bruce Johnson <johnson@coral.rice.edu>
Subject: Re: Triadic wavelets---is there such a thing?
Date: 3 Jan 2000 10:00:02 -0600
Newsgroups: sci.math.research
Summary: [missing]

Uday Patil <udaypatil@home.com> wrote:
> Has anybody extended the idea of a dyadic-wavelet 
> ( f_i(x*2^n) n = 0,1,2,...   forming a basis) 
> to triadic or other scaling schemes?
> ( f_i(x*3^n)  or f_i(x*k^n) forming a basis)

Yes, these come under the heading of M-band wavelets (M=2 being
the ordinary case).  You can have orthogonal compact support
systems provided M is an integer.  Check under www.wavelet.org
for various wavelet links that will have M-band material.

> If so, what is the triadic equivalent of the 
> dyadic, Haar basis?

The usual formulation is that you have one scaling function and
M-1 wavelets (2 of them for the M=3 case).  I have not checked
for a Haar analog, but one natural choice of basis is probably
piecewise constant functions on (0,1),(1,2),(2,3) with the values
phi:  (1,1,1)/Sqrt[3]
psi1: (-1,2,-1)/Sqrt[6]
psi2: (-1,0,1)/Sqrt[2]
One can perform any orthogonal transformation in the (psi1,psi2)
space to generate different bases, too.

Bruce