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This book contains the refereed papers which were presented at the
second In ternational Dortmund Meeting on Approximation Theory
(IDoMAT 98) at Haus Bommerholz, the conference center of Dortmund
University, during the week of February 23-27,1998. At this
conference 52 researchers and specialists from Bul garia, China,
France, Great Britain, Hungary, Israel, Italy, Roumania, South
Africa and Germany participated and described new developments in
the fields of uni variate and multivariate approximation theory.
The papers cover topics such as radial basis functions, bivariate
spline interpolation, multilevel interpolation, mul tivariate
triangular Bernstein bases, Pade approximation, comonotone
polynomial approximation, weighted and unweighted polynomial
approximation, adaptive ap proximation, approximation operators of
binomial type, quasi interpolants, gen eralized convexity and Peano
kernel techniques. This research has applications in areas such as
computer aided geometric design, as applied in engineering and
medical technology (e. g. computerised tomography). Again this
international conference was wholly organized by the Dortmund
Lehrstuhl VIII for Approximation Theory. The organizers attached
great impor tance to inviting not only well-known researchers but
also young talented math ematicians. IDoMAT 98 gave an excellent
opportunity for talks and discussions between researchers from
different fields of Approximation Theory. In this way the
conference was characterized by a warm and cordial atmosphere. The
success of IDoMAT 98 was above all due to everyone of the
participants."
The current form of modern approximation theory is shaped by many
new de velopments which are the subject of this series of
conferences. The International Meetings on Approximation Theory
attempt to keep track in particular of fun damental advances in the
theory of function approximation, for example by (or thogonal)
polynomials, (weighted) interpolation, multivariate
quasi-interpolation, splines, radial basis functions and several
others. This includes both approxima tion order and error
estimates, as well as constructions of function systems for
approximation of functions on Euclidean spaces and spheres. It is a
piece of very good fortune that at all of the IDoMAT meetings, col
leagues and friends from all over Europe, and indeed some count
ries outside Europe and as far away as China, New Zealand, South
Africa and U.S.A. came and dis cussed mathematics at IDoMAT
conference facility in Witten-Bommerholz. The conference was, as
always, held in a friendly and congenial atmosphere. After each
meeting, the delegat es were invited to contribute to the proceed
ing's volume, the previous one being published in the same
Birkhauser series as this one. The editors were pleased about the
quality of the contributions which could be solicited for the book.
They are refereed and we should mention our gratitude to the
referees and their work."
In many areas of mathematics, science and engineering, from
computer graphics to inverse methods to signal processing, it is
necessary to estimate parameters, usually multidimensional, by
approximation and interpolation. Radial basis functions are a
powerful tool which work well in very general circumstances and so
are becoming of widespread use as the limitations of other methods,
such as least squares, polynomial interpolation or wavelet-based,
become apparent. The author's aim is to give a thorough treatment
from both the theoretical and practical implementation viewpoints.
For example, he emphasises the many positive features of radial
basis functions such as the unique solvability of the interpolation
problem, the computation of interpolants, their smoothness and
convergence and provides a careful classification of the radial
basis functions into types that have different convergence. A
comprehensive bibliography rounds off what will prove a very
valuable work.
It is necessary to estimate parameters by approximation and interpolation in many areas-from computer graphics to inverse methods to signal processing. Radial basis functions are modern, powerful tools which are being used more widely as the limitations of other methods become apparent. Martin Buhmann provides a complete analysis of radial basic functions from the theoretical and practical implementation viewpoints. He also includes a comprehensive bibliography.
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