BackgroundA previous question of mine lead me up to the concept of Kravchuk polynomials for a minimal basis of three polynomials of order 1,2 and 3.
Simultaneously in the multiresolution analysis of discrete wavelet transforms (DWT), we have the concept of orthogonal and orthonormal wavelet bases. These carry the property that the scalar products between the low pass and high pass is zero.
The Krawtchuk decomposition above allows us to define local ON bases so that in the same sense as for DWT above. The polynomials of order > 0 will be forced to describe wavelets in the same sense as in the ON DWTs as the involutional property forces all the scalar products with higher order polynomials to be 0.
Now this will so far only allow construction of disjoint multiresolution transforms. (In other words : no "overlap" of the filters). This greatly relaxes and simplifies the otherwise quite tedious relations of designing DWTs. The in general quite hard problem of polynomial factorization.
As a minimal example we can consider a vector of $N$ floating point values ${\bf v}^{N\times 1}$, we can call it a signal. Now we can with Kronecker products construct a first level analysis by a block-diagonal matrix followed by a permutation moving all the 0 mod 3 positions first, followed by all the 1 mod 3 positions and lastly all the 2 mod 3 positions. This will allow a "fast transform" and iteration to be possible on the low-pass channel only as the cascade algorithm, or on any number of them as per the wavelet-packet analysis.
One big difference is that for a N polynomial Krawchuck-basis for critical sampling we will have a subsampling of N after each new transform level. In our case with N=3 using the cascade algorithm, the low pass coefficients will be reduced to one third after each step and not one half.
Experiment with DWT filters in old question. Each level has 1 low pass band and 2 high pass bands, giving not dyadic subdivision but rather triadic (?)
Now to the question : how can this be generalized with over-lapping filters but still keeping a polynomial basis? Can we modify the Krawchuk polynomials to stay orthogonal also with dilated versions? Or if we can add extra conditions for the design of ON DWTs which impose the Krawchukian nature of the frequency bands?