Interpolating Composite Systems

Using recent results where we constructed refinable functions for composite dilations, we construct interpolating multiscale decompositions for such systems. Interpolating wavelets are a well-known construction similar to the usual L2-theory but with L2-projectors onto the scaling spaces replaced by L∞projectors defined by an interpolation procedure. A remarkable result is that this much simpler transform satisfies the same norm-equivalences as L2-wavelets between Besov-space (or Triebel-Lizorskin-space) norms and discrete norms on the coefficients – provided the space embeds into L∞, see [4]. Since the interpolating wavelet transform is not stable in an L2-sense, and since there exist many nice and general L2 wavelet constructions [2], usually the more complicated L2-theory is preferred (although L2-stability is by no means necessary for many applications like for instance pde-solver [9]). Nevertheless there exist situations to which the interpolating wavelet construction can be generalized, whereas the generalization of the L2-constructions is much more difficult or even impossible. One such case is that of manifold-valued data where interpolating wavelets can be defined and it can also be shown that they satisfy the same desirable properties as their linear counterparts [13, 7]. On the other hand, the scope of L2-wavelet constructions which are amemable to generalization to manifoldvalued data is very limited, see [6]. Another such case is if the domain of the data is a manifold. Then one can first build a hierachical triangulation of the manifold and define interpolating wavelets in a straightforward way. By using the general method of lifting, see [14], it is then possible to even generate L2-stable MRAs from this initial simple multiscale decomposition. Philipp Grohs Philipp Grohs, TU Graz, Institute of Geometry, e-mail: [email protected]