Spline Functions and Multivariate Interpolations
Spline Functions and Multivariate Interpolations

Author: Borislav D. Bojanov

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 287

ISBN-13: 940158169X

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Spline functions entered Approximation Theory as solutions of natural extremal problems. A typical example is the problem of drawing a function curve through given n + k points that has a minimal norm of its k-th derivative. Isolated facts about the functions, now called splines, can be found in the papers of L. Euler, A. Lebesgue, G. Birkhoff, J. Favard, L. Tschakaloff. However, the Theory of Spline Functions has developed in the last 30 years by the effort of dozens of mathematicians. Recent fundamental results on multivariate polynomial interpolation and multivari ate splines have initiated a new wave of theoretical investigations and variety of applications. The purpose of this book is to introduce the reader to the theory of spline functions. The emphasis is given to some new developments, such as the general Birkoff's type interpolation, the extremal properties of the splines and their prominant role in the optimal recovery of functions, multivariate interpolation by polynomials and splines. The material presented is based on the lectures of the authors, given to the students at the University of Sofia and Yerevan University during the last 10 years. Some more elementary results are left as excercises and detailed hints are given.

Multivariate Splines
Multivariate Splines

Author: Charles K. Chui

Publisher: SIAM

Published: 1988-01-01

Total Pages: 192

ISBN-13: 0898712262

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Subject of multivariate splines presented from an elementary point of view; includes many open problems.

Interpolation and Approximation with Splines and Fractals
Interpolation and Approximation with Splines and Fractals

Author: Peter Robert Massopust

Publisher:

Published: 2010

Total Pages: 344

ISBN-13:

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This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and fractal surfaces. This synthesis will complement currently required courses dealing with these topics and expose the prospective reader to some new and deep relationships. In addition to providing a classical introduction to the main issues involving approximation and interpolation with uni- and multivariate splines, cardinal and exponential splines, and their connection to wavelets and multiscale analysis, which comprises the first half of the book, the second half will describe fractals, fractal functions and fractal surfaces, and their properties. This also includes the new burgeoning theory of superfractals and superfractal functions. The theory of splines is well-established but the relationship to fractal functions is novel. Throughout the book, connections between these two apparently different areas will be exposed and presented. In this way, more options are given to the prospective reader who will encounter complex approximation and interpolation problems in real-world modeling. Numerous examples, figures, and exercises accompany the material.

Multivariate Spline Functions and Their Applications
Multivariate Spline Functions and Their Applications

Author: Ren-Hong Wang

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 522

ISBN-13: 9401723788

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This book deals with the algebraic geometric method of studying multivariate splines. Topics treated include: the theory of multivariate spline spaces, higher-dimensional splines, rational splines, piecewise algebraic variety (including piecewise algebraic curves and surfaces) and applications in the finite element method and computer-aided geometric design. Many new results are given. Audience: This volume will be of interest to researchers and graduate students whose work involves approximations and expansions, numerical analysis, computational geometry, image processing and CAD/CAM.

Interpolation and Approximation by Polynomials
Interpolation and Approximation by Polynomials

Author: George M. Phillips

Publisher: Springer Science & Business Media

Published: 2006-04-06

Total Pages: 325

ISBN-13: 0387216820

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In addition to coverage of univariate interpolation and approximation, the text includes material on multivariate interpolation and multivariate numerical integration, a generalization of the Bernstein polynomials that has not previously appeared in book form, and a greater coverage of Peano kernel theory than is found in most textbooks. There are many worked examples and each section ends with a number of carefully selected problems that extend the student's understanding of the text. The author is well known for his clarity of writing and his many contributions as a researcher in approximation theory.

Multivariate Approximation and Splines
Multivariate Approximation and Splines

Author: Günther Nürnberger

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 329

ISBN-13: 3034888716

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This book contains the refereed papers which were presented at the interna tional conference on "Multivariate Approximation and Splines" held in Mannheim, Germany, on September 7-10,1996. Fifty experts from Bulgaria, England, France, Israel, Netherlands, Norway, Poland, Switzerland, Ukraine, USA and Germany participated in the symposium. It was the aim of the conference to give an overview of recent developments in multivariate approximation with special emphasis on spline methods. The field is characterized by rapidly developing branches such as approximation, data fit ting, interpolation, splines, radial basis functions, neural networks, computer aided design methods, subdivision algorithms and wavelets. The research has applications in areas like industrial production, visualization, pattern recognition, image and signal processing, cognitive systems and modeling in geology, physics, biology and medicine. In the following, we briefly describe the contents of the papers. Exact inequalities of Kolmogorov type which estimate the derivatives of mul the paper of BABENKO, KOFANovand tivariate periodic functions are derived in PICHUGOV. These inequalities are applied to the approximation of classes of mul tivariate periodic functions and to the approximation by quasi-polynomials. BAINOV, DISHLIEV and HRISTOVA investigate initial value problems for non linear impulse differential-difference equations which have many applications in simulating real processes. By applying iterative techniques, sequences of lower and upper solutions are constructed which converge to a solution of the initial value problem.

The Theory of Splines and Their Applications
The Theory of Splines and Their Applications

Author: J. H. Ahlberg

Publisher: Elsevier

Published: 2016-06-03

Total Pages: 297

ISBN-13: 1483222950

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The Theory of Splines and Their Applications discusses spline theory, the theory of cubic splines, polynomial splines of higher degree, generalized splines, doubly cubic splines, and two-dimensional generalized splines. The book explains the equations of the spline, procedures for applications of the spline, convergence properties, equal-interval splines, and special formulas for numerical differentiation or integration. The text explores the intrinsic properties of cubic splines including the Hilbert space interpretation, transformations defined by a mesh, and some connections with space technology concerning the payload of a rocket. The book also discusses the theory of polynomial splines of odd degree which can be approached through algebraically (which depends primarily on the examination in detail of the linear system of equations defining the spline). The theory can also be approached intrinsically (which exploits the consequences of basic integral relations existing between functions and approximating spline functions). The text also considers the second integral relation, raising the order of convergence, and the limits on the order of convergence. The book will prove useful for mathematicians, physicist, engineers, or academicians in the field of technology and applied mathematics.

Spline Functions: Basic Theory
Spline Functions: Basic Theory

Author: Larry Schumaker

Publisher: Cambridge University Press

Published: 2007-08-16

Total Pages: 524

ISBN-13: 1139463438

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This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.

Approximation Theory and Spline Functions
Approximation Theory and Spline Functions

Author: S.P. Singh

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 481

ISBN-13: 9400964668

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A NATO Advanced Study Institute on Approximation Theory and Spline Functions was held at Memorial University of Newfoundland during August 22-September 2, 1983. This volume consists of the Proceedings of that Institute. These Proceedings include the main invited talks and contributed papers given during the Institute. The aim of these lectures was to bring together Mathematicians, Physicists and Engineers working in the field. The lectures covered a wide range including ~1ultivariate Approximation, Spline Functions, Rational Approximation, Applications of Elliptic Integrals and Functions in the Theory of Approximation, and Pade Approximation. We express our sincere thanks to Professors E. W. Cheney, J. Meinguet, J. M. Phillips and H. Werner, members of the International Advisory Committee. We also extend our thanks to the main speakers and the invi ted speakers, whose contri butions made these Proceedings complete. The Advanced Study Institute was financed by the NATO Scientific Affairs Division. We express our thanks for the generous support. We wish to thank members of the Department of Mathematics and Statistics at MeMorial University who willingly helped with the planning and organizing of the Institute. Special thanks go to Mrs. Mary Pike who helped immensely in the planning and organizing of the Institute, and to Miss Rosalind Genge for her careful and excellent typing of the manuscript of these Proceedings.