Multidimensional minimizing splines. Theory and applications.(English)Zbl 1064.41001

Boston, MA: Kluwer Academic Publishers (ISBN 1-4020-7786-6/hbk). xvi, 261 p. (2004).
This book is devoted to the theory of multivariate minimizing splines and some applications in surface approximation. It is mainly written for postgraduates. If a quadratic functional involving derivatives of order $$m$$ is minimized in a Hilbert space $$E$$, then one obtains minimizing splines, in particular multivariate $$D^m$$-splines introduced by J. Duchon [RAIRO, Anal. Numér. 12, 325–334 (1978; Zbl 0403.41003)]. If $$E$$ is a Sobolev-type space over $${\mathbb R}^n$$ with smoothness order $$s\in (-m+n/2,\,n/2)$$, then one gets $$(m,s)$$-splines or polyharmonic splines. The $$(m,0)$$-splines are the $$D^m$$-splines over $${\mathbb R}^n$$. For $$n=m=2$$, one obtains the thin plate splines.
This well-written book is divided into 3 parts. Part A presents the theory of $$(m,s)$$-splines over $${\mathbb R}^n$$. Since the authors do not use reproducing kernels, the approach presented in this book allows the rediscovery of Duchon’s results and the proof of new results concerning the convergence and approximation error. In Part B, the authors treat $$D^m$$-splines over a bounded domain $$\Omega\subset {\mathbb R}^n$$. Further, discrete $$D^m$$-splines are introduced as approximants of finite element type of the $$D^m$$-splines over $$\Omega$$. Finally, Part C discusses some applications in geophysics and geology arising in oil research. In each example, the surface approximation is based on discrete $$D^m$$-splines.

MSC:

 41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 41A63 Multidimensional problems 65D07 Numerical computation using splines 65D17 Computer-aided design (modeling of curves and surfaces)

Zbl 0403.41003