Abstract

We introduce the concept of samplets, extending the Tausch-White multi-wavelet construction to scattered data. This results in a multiresolution analysis of discrete signed measures with vanishing moments, enabling efficient data compression, feature detection, and adaptivity. The cost for constructing the samplet basis and applying the fast samplet transform is linear in the number of data sites N. We apply samplets to compress kernel matrices for scattered data approximation, achieving sparse matrices with only O(N log N) non-zero entries in the case of quasi-uniform data. The approximation error is controlled by the number of vanishing moments.

As application, we present multiscale interpolation schemes for scattered data using globally supported radial basis functions. This method constructs a multiscale approximation through a sequence of residual corrections, where radial basis functions with varying length-scale parameters capture different levels of detail. To efficiently handle large datasets, we represent the resulting generalized Vandermonde matrices in samplet coordinates, enabling the interpolation of millions of points. As further application, we investigate smoothness class detection using samplets. By analyzing the samplet transform of a data set, we identify regions in the domain exhibiting singular behavior and classify their nature by examining the decay rate of coefficients across different levels of the multiresolution analysis.