This book is a collection of eleven articles, written by leading experts and dealing with special topics in Multivariate Approximation and Interpolation. The material discussed here has far-reaching applications in many areas of Applied Mathematics, such as in Computer Aided Geometric Design, in Mathematical Modelling, in Signal and Image Processing and in Machine Learning, to mention a few. The book aims at giving a comprehensive information leading the reader from the fundamental notions and results of each field to the forefront of research. It is an ideal and up-to-date introduction for graduate students specializing in these topics, and for researchers in universities and in industry. - A collection of articles of highest scientific standard - An excellent introduction and overview of recent topics from multivariate approximation - A valuable source of references for specialists in the field - A representation of the state-of-the-art in selected areas of multivariate approximation - A rigorous mathematical introduction to special topics of interdisciplinary research

Topics in Multivariate Approximation contains the proceedings of an international workshop on multivariate approximation held at the University of Chile in Santiago, Chile, on December 15-19, 1986. Leading researchers in the field discussed several problem areas related to multivariate approximation and tackled topics ranging from multivariate splines and fitting of scattered data to tensor approximation methods and multivariate polynomial approximation. Numerical grid generation and finite element methods were also explored, along with constrained interpolation and smoothing. Comprised of 22 chapters, this book first describes the application of Boolean methods of approximation in combination with the theory of right invertible operators to bivariate Fourier expansions. The reader is then introduced to ill-posed problems in multivariate approximation; interpolation of scattered data by radial functions; and shape-preserving surface interpolation. Subsequent chapters focus on approximation by harmonic functions; numerical generation of nested series of general triangular grids; triangulation methods; and inequalities arising from best local approximations in rectangles. A bibliography of multivariate approximation concludes the book. This monograph will be of interest to mathematicians.

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.

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.

This volume consists of 24 refereed carefully edited papers on various topics in multivariate approximation. It represents the proceedings of a workshop organized by the University of Firenze, and held in September 1995 in Montecatini, Italy.The main themes of the volume are multiresolution analysis and wavelets, multidimensional interpolation and smoothing, and computer-aided geometric design. A number of particular topics are included, like subdivision algorithms, constrained approximation and shape-preserving algorithms, thin plate splines, radial basis functions, treatment of scattered data, rational surfaces and offsets, blossoming, grid generation, surface reconstruction, algebraic curves and surfaces, and neural networks.

Self-contained presentation of multivariate approximation from classical linear approximation to contemporary nonlinear approximation.

This book introduces general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators. On this background, the book builds the first comprehensive introduction to the theory of generalized hyperinterpolation. Several parts of the book are based on rotation principles, which are presented in the beginning of the book.

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.

Approximation theory in the multivariate setting has many applications including numerical analysis, wavelet analysis, signal processing, geographic information systems, computer aided geometric design and computer graphics. This advanced introduction to multivariate approximation and related topics consists of nine articles written by leading experts surveying many of the new ideas and their applications. Each article takes the reader to the forefront of research and ends with a comprehensive bibliography.

Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to support courses in computer-aided geometric design, and meshless methods for partial differential equations.