Appendix A: Higher Multiplier Constructions

Peter Heller has extensively developed the multiplier m wavelet transform [5],

It is straightforward to apply our invariant basis constructions in the higher multiplier case. The identity for multiplier m and n=km

relates shifts of size m to simple (unit) shifts on the m components of the wavelet transform. Therefore, shifts of order lm of the data produce order l shifts in the wavelet transform components. This identity is a direct consequence of the identity

where the coefficientsdefine the scaling function by the scaling relation

To define a order m g-invariant transform based on the g-circulant (order p) group, we form the set of mp transformations

and calculate for data

for. The g-invariant transform is

where

for an additive information cost function, M. To complete the transform when the minimum is not unique we proceed as before to otherwise distinguish the set of. The g-invariance is a direct consequence of the above identity for the wavelet transform of m-shifts.