Samplets: Construction and scattered data compression

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. The result is a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity.
The cost for constructing the samplet basis and for the fast samplet transform, respectively, are linear in the size N of the data set. We employ samplets with vanishing moments to compress kernel matrices for efficient scattered data approximation. The compressed matrices are sparse and have only O(N log N) entries.
The entailed approximation error is controllable by the number of vanishing moments. We finish the presententation with numerical studies for scattered data approximation with sparsity constraints.