Multivariate Spline Functions and Their Applications

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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 1 above with many changes, including that of the structure of a cubic - interpolation in n-dimensional spline spaces, and more detail on triangu­ lations have been added in this book. Special cases of multivariate spline functions (such as step functions, polygonal functions, and piecewise polynomials) have been examined math­ ematically 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 Interpo­ lations theories der reellen periodischen function, Press. Akad. Wiss. (PhysMath. KL), 30(1938), 383- 429) systematically established the the­ ory of the spline functions. W. Quade & L. Collatz mainly discussed the periodic functions, while I. J. Schoenberg's work was systematic and com­ plete. I. J. Schoenberg outlined three viewpoints for studing univariate splines: Fourier transformations, truncated polynomials and Taylor ex­ pansions. Based on the first two viewpoints, I. J. Schoenberg deduced the B-spline function and its basic properties, especially the basis func­ tions. 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.