Notes from Leo's visit to UMSL in June 05.

Properties desired

What properties are easiest (easy to hard)?

1. polynomial reproduction
2. refinement
3. vanishing moments
4. compact support

Subspaces Vi subset Vi+1, with each space defined by basis.

Wavelet question is about constructing a hierarchy of spaces defined by basis functions that have refinement and vanishing moments. Defined by knot insertion and removal (with arbitrary spacing, can insert however you want.)

What properties are important (most to least)?

1. refinement
2. polynomial reproduction
3. vanishing moments
4. compact support

Refinement and polynomial reproduction

Wenjie and Charles think we should start with linear case and try to do all of the above. First focus on first two.

Refinement is harder in 2d with the simplex spline.

We have a sequence of subspaces Vi subset Vi+1, with each space Vi defined as a linear combination of elements of a basis Bi. Refinement means that in going from V_i to Vi+1, which means going from Bi to Bi+1, you need to be able to reproduce the old vector space Vi with the new basis Bi+1. In 2d, this is not possible using Delaunay...is it ever possible with irregular points?

(Leo wonders about leaving old things there. May introduce some dependencies, but should never be too large.)

Attempt(s)

Start with triangulation and let basis function be "Courant elements" which are stars of vertices with cone interp. Try doing this with simplex splines -- you know you have polynomial reproduction if you can reproduce the Courant elements. Problem with refinement, however. (spell out?)

Neamtu's basis: he actually proposes two.

1. k-Delaunay configuration defined by circles containing k sites.
2. Group all k-Delaunay that share the same interior points. (More stable, also defn more stable -- doesn't need general position.)

(Leo asks: Does 2. help refinement? Reproduce star instead of reproducing the individual triangles that make it up; cuts both ways, though.)

Wenjie and Charles are...

Looking for a definition, not based on Delaunay, for 1st degree basis. Approach is to choose a basis from a triangulation, and modify it by modifing the triangulation.

Random notes to be expanded.

NB: no papers with wavelets on TIN from mathematician's point of view. (what is the abstract of a paper that would satisfy Charles?)

-- JackSnoeyink - 30 Jun 2005