Samplets: Wavelet Concepts for Scattered Data

This chapter is dedicated to recent developments in the field of wavelet analysis for scattered data. We introduce the concept of samplets, which are signed measures of wavelet type and may be defined on sets of arbitrarily distributed data sites in possibly high dimension. By employing samplets, we transfer well-known concepts known from wavelet analysis, namely the fast basis transform, data compression, operator compression and operator arithmetics to scattered data problems. Especially, samplet matrix compression facilitates the rapid solution of scattered data interpolation problems, even for kernel functions with nonlocal support. Finally, we demonstrate that sparsity constraints for scattered data approximation problems become meaningful and can efficiently be solved in samplet coordinates.