digression: eigenvalue 1
Before I proceed with consequences of orthogonality, I need to mention an omission. For one thing, I have gotten so caught up in the properties I’ve been looking at, that I have forgotten one of the crucial ones we used earlier. For another thing, the consequence which I have forgotten is still a little strange to me.
The consequence (or consequences)?
- That the dilation equation could be written as an eigenvalue equation,
- that the existence of a scaling function seems to be equivalent to an eigenvalue = 1,
- and that the values of the scaling function at the integers are given by the corresponding eigenvector.
This has been crucial to some of our work: the recursion which I use for computing the scaling function is initialized by setting the values of the scaling function at the integers — that is, by finding the eigenvector with eigenvalue 1. Recursion — especially when combined with a lookup table — is very easy and very powerful; but we absolutely had to have initial values, and that eigenvector provided them. (I did this for the Daubechies D4, and for four wavelet systems with 6 nonzero h’s.)
The strangeness? Read the rest of this entry »