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Commutation for irregular subdivision. (English) Zbl 0994.42019
In the paper, subdivision schemes on multilevel grids on the real line are considered. The grids are assumed to be two-nested or two-threadable. The subdivision scheme is a pair \((S,X)\) where \(S\) is a sequence of linear operators \(S_j\), \(j\geq 0\), connected with the multilevel grid \(X\).
In the regular case, the subdivision scheme satisfies a so-called commutation property. That is, starting with two biorthogonal subdivision schemes \(S,\widetilde{S}\), application of the forward and backward difference operator, \(( \Delta a)_k=a_{k+1}-a_k, (\widetilde\Delta a)_k=a_k-a_{k-1}\), provides two new biorthogonal subdivision schemes \(S^1, \widetilde S^1\) by \[ 2\Delta S=S^1\Delta\quad\text{and}\quad 2\widetilde\Delta\widetilde S^1=\widetilde S\widetilde\Delta. \] In the paper this property is transferred to irregular subdivision. For that purpose one needs to define divided differences adapted to the irregular subdivision scheme.
A generalization of the commutation property and an inverse commutation property are discussed.
It is shown, how to construct wavelets associated with these subdivision schemes. The constructions are illustrated by Lagrange interpolation subdivision and by B-spline schemes.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15 Spline approximation
65D07 Numerical computation using splines
47B39 Linear difference operators
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