Scaling function and wavelet functions are now vectors -> Coefficients become matrix (2d) instead of list
Paper “Adaptive Solution of Partial Differential Equations in Multiwavelet Bases”1:
- Constructed on bounded interval (here [0,1])
- Proposes an interpolating basis (-> Coefficients are values) with Lagrange interpolating polynomials
- Results in basis functions $R_j(x) = \frac{1}{\sqrt{w_j}} l_j(x)$ with $l_j$ being the Lagrange polynomials
- Opposite way: Scaling and wavelet functions are known, need to construct the coefficients (not in our case?)
- -> Explicit expression for scaling function! (although a bit complicated)
- Not sure about the exact properties, seem to be few coefficients on coarsest level
- Explicit section about (first) derivative -> is not unique for discontinuous functions?
Interesting family suggested by Markus Bachmayr: Multiwavelets developed by Donovan, Geronimo, Hardin