Multivariate Spline Functions and Their Applications（英文影印版）（硬皮精装）

　　this book deals with the algebraic geometric method of studying multivariate splines. topics treated include: the theory of multivariate spline spaces, higher dimensional spline, rational spline, piecewise algebraic variety (including piecewise algebraic curve and surface) and applications in the finite element method and computer aided geometric design. many new results are given. .
　　 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. ...

chapter 1 introduction to multivariate spline functions
1.1 basic frame of multivariate spline functions.
1.2 generalized truncated power function and general representation of multivariate spline functions
1.3 interpolation by multivariate spline functions
1.4 weighted spline and splines with different smoothness
1.5 introduction to multivariate rational splines
1.6 n-dimensional spline functions
chapter 2 multivariate spline spaces
2.1 multivariate spline spaces on cross-cut partitions
2.2 spline spaces on rectangular and simple cross-cut partitions
2.3 spline spaces on type-1 triangulations
2.4 spline spaces on type-2 triangulations
2.5 spline spaces on some non-uniform triangulations
2.6 spline spaces with boundary conditions on uniform type-1 and type-2 triangulations
2.7 spline spaces with boundary conditions on non-uniform type-2 triangulations
2.8 dimension of spline space skμ(δ) on triangulations
2.9 decomposition method for studying multivariate splines
2.9.1 examples
chapter 3 other methods for studying multivariate spline functions
3.1 b-spline method

　　As is known, the book named "Multivariate spline functions and their applications" has been published by the Science Press in 1994. .
　　This book is an English edition based on the original book mentioned above with many changes, including that of the structure of a cubic C1-interpolation in n-dimensional spline spaces, and more detail on triangulations have been added in this book.
　　Special cases of multivariate spline functions (such as step functions, polygonal functions, and piecewise polynomials) have been examined mathematically for a long time. I. J. Schoenberg (Contribution to the problem of application of equidistant data by analytic functions, Quart. Appl. Math., 4(1946),45 - 99; 112 - 141) and W. Quade & L. Collatz (Zur Interpolations theories der reeUen periodischen function, Press. Akad. Wiss. (PhysMath. KL), 30(1938), 383 - 429) systematically established the theory of the spline functions. W. Quade & L. Collatz mainly discussed the periodic functions, while I. J. Schoenberg's work was systematic and complete. I. J. Schoenberg outlined three viewpoints for studing univariate splines: Fourier transformations, truncated polynomials and Taylor expansions. Based on the first two viewpoints, I. J. Schoenberg deduced the B-spline function and its basic properties, especially the basis functions. Based on the latter viewpoint, he represented the spline functions in terms of truncated polynomials. These viewpoints and methods had significantly effected on the development of the spline functions.
　　In view of the variety and complexity in application, it is very important to study the multivariate spline function theoretically. Since the multivariate spline function is heavily dependent on the geometric property of the domain partitions, it is so complex that the multivariate spline function, especially the non-Cartesian product multivariate spline function, has not been developed radically for a long time. G. Birkhoff, H. L. Garabedian, C. de Boor, M. H. Schultz and R. S. Varga discussed the Cartesian product bicubic spline function and its applications in numerical solutions of partial differential equations. ..
　　Analysing the relation between the polynomials over two adjacent cells, we introduce the smooth cofactor and conformality condition to which the polynomials must satisfy. The conformality condition establishes the equivalent conversion between the multivariate spline function and the corresponding algebraic problem. Moreover, the conformality condition provides an algebraic approach to studying the multivariate spline function. Based on the conformality condition theory, we have systematically studied the dimension of the multivariate spline functions, the basis functions, especially the locally supported basis functions, the smooth surface interpolations, the non-linear spline interpolations, the higher-dimensional spline functions, and the multivariate spline functions in computer aided geometric designs.
　　This book will systematically introduce the basic theories and methods on the multivariate spline functions. In order for the reader to know the frontier research on the multivariate spline functions, we will also introduce the modern developments of the multivariate spline functions and their applications in sciences and engineering. More precisely, Chapter 1 introduces the basic definitions of the multivariate spline functions, facts, and results; Chapter 2 mainly introduces the dimension of the multivariate spline function space the theory on the basis functions, and their constructions; Chapter 3 mainly introduces the notable Box spline, the simplex spline, and the B-net method, etc.; Chapter 4 introduces the basic theory, methods, and structures of the higher-dimensional spline functions; Chapter 5 introduces the theory on non-linear spline interpolations and their constructive methods; Chapter 6 introduces the basic problems and results on the piecewise algebraic curves and the piecewise algebraic surfaces; Chapter 7 introduces applications of the multivariate spline functions in the sciences and engineering, especially in finite element methods and computer aided geometric designs.
　　The writing of this book was participated in by professors Xiquan Shi, Zhongxuan Luo, Zhixun Su, and Dr. Shao-Ming Wang who is also the translator of this book. I wish to express my great appreciation to the Publishing Foundation of Academia Sinica, as will as The National Nature Science Foundation of China. Without their assistance, this book is unable to be published. ...
　　Ren-Hong Wang
　　Institute of Applied Mathematics
　　Dalian University of Technology
　　Dalian, P.R.China
　　October, 2000