Box splines
Author(s)
Bibliographic Information
Box splines
（Applied mathematical sciences, v. 98）
Springer, c1993
- : us
- : gw
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Note
Includes bibliographical references (p. 175-192) and index
Description and Table of Contents
- Volume
-
: us ISBN 9780387941011
Description
Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.
Table of Contents
I * Box splines defined.- II * The linear algebra of box spline spaces.- III * Quasi-interpolants & approximation power.- IV * Cardinal interpolation & difference equations.- V * Approximation by cardinal splines & wavelets.- VI * Discrete box splines & linear diophantine equations.- VII Subdivision algorithms.- References.
- Volume
-
: gw ISBN 9783540941019
Description
This text on box splines provides a complete treatment for any kind of multivariate spline. Box splines give rise to an intriguing and beautiful mathematical theory that is much richer and more intricate than the univariate case because of the complexity of smoothly-joining polynomial pieces on polyhedral cells. The purpose of this book is to provide the basic facts about box splines in a cohesive way with simple, complete proofs, many illustrations, and with an up-to-date bibliography. It is not the book's intention to be encyclopaedic about the subject, but rather to provide the fundamental knowledge necessary to familiarize graduate students and researchers in analysis, numerical analysis and engineering with a subject that surely will have as many widespread applications as its univariate predecessor. The book begins with chapters on box splines defined, linear algebra of box spline spaces, and quasi-interpolants and approximation power. It continues with cardinal interpolation and difference equations, and approximation by cardinal splines and wavelets. The book concludes with discrete box splines and linear diophantine equations, and subdivision algorithms.
by "Nielsen BookData"
