caa a22 4500 467939349 CHVBK 20180406153007.0 cr unu---uuuuu 170328e20061101xx s 000 0 eng 10.1007/s10444-004-7622-3 doi (NATIONALLICENCE)springer-10.1007/s10444-004-7622-3 Opfer Roland Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestraße 16-18, 37073, Göttingen, Germany aut Multiscale kernels [Elektronische Daten] [Roland Opfer] This paper reconstructs multivariate functions from scattered data by a new multiscale technique. The reconstruction uses standard methods of interpolation by positive definite reproducing kernels in Hilbert spaces. But it adopts techniques from wavelet theory and shift-invariant spaces to construct a new class of kernels as multiscale superpositions of shifts and scales of a single compactly supported function φ. This means that the advantages of scaled regular grids are used to construct the kernels, while the advantages of unrestricted scattered data interpolation are maintained after the kernels are constructed. Using such a multiscale kernel, the reconstruction method interpolates at given scattered data. No manipulations of the data (e.g., thinning or separation into subsets of certain scales) are needed. Then, the multiscale structure of the kernel allows to represent the interpolant on regular grids on all scales involved, with cheap evaluation due to the compact support of the function φ, and with a recursive evaluation technique if φ is chosen to be refinable. There also is a wavelet-like data reduction effect, if a suitable thresholding strategy is applied to the coefficients of the interpolant when represented over a scaled grid. Various numerical examples are presented, illustrating the multiresolution and data compression effects. Springer, 2006 scattered data approximation nationallicence radial basis functions nationallicence wavelets nationallicence refineable functions nationallicence multiresolution nationallicence data compression nationallicence Advances in Computational Mathematics Springer Netherlands 25/4(2006-11-01), 357-380 1019-7168 25:4<357 2006 25 10444 https://doi.org/10.1007/s10444-004-7622-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10444-004-7622-3 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Opfer Roland Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestraße 16-18, 37073, Göttingen, Germany aut NATIONALLICENCE 773 0- Advances in Computational Mathematics Springer Netherlands 25/4(2006-11-01), 357-380 1019-7168 25:4<357 2006 25 10444 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer