Wavelets on manifolds: An optimized construction

A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds from [DS], which is based on topological isomorphisms [CF], is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required in [DS]. In the second part of the paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from [DS]. As an example, we illustrate a wavelet basis on the 2–sphere.