Multivariate Spline Approximation
Final rept. 27 Sep 1990-26 Sep 1993
WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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We started out with the goal of understanding approximation order in a multivariate context, including the approximation of surfaces. In addition, we wanted to understand better the use and analysis of our approach to multivariate polynomial interpolation. We ended up concentrating on approximation from shift- invariant spaces of functions on IR real space. Here, S is shift-invariant if f is an element S implies that also f. - alpha is an element S for any integer vector alpha. The simple, yet widely applicable, model we considered concerns the behavior, as H approaches 0, of the distance between f and s of a suitably smooth f from the scaled space S sub h f.h f is an element S sub h, with each S sub h a shift-invariant space. Examples of such spaces are provided by finite elements on a regular grid, in particular, box spline spaces, also the spaces which make up the multiresolution analysis generated by wavelets.
- Numerical Mathematics
- Statistics and Probability