B样条函数：历史发展与应用

多元B样条的发展可以说是历史上激动人心的篇章.有很多关于B样条里程碑式的研究，甚至一些统计学的论文中也有B样条理论的历史渊源.I.J.Schoenberg被称为样条函数之父，他推动了样条函数的理论发展.Schoenberg在1956年5月31日写给P.J.Davis的信件中描述了他关于二元B样条尚未发表的一些想法.为了给读者再现这段历史，我们将这封信的原文附上.在一篇似乎不相关的论文中，Motzkin和Schoenberg[92]研究了一类多元整函数.在一元的情况中，这类函数是Curry和Schoenberg研究B样条函数极限分布时用到的重要工具.在文献[80]中，我们可以查到这些思想.此后，在文献[93]中，我们能够将Motzkin-Schoenberg函数和这类作为多元B样条函数的极限形式的函数联系在一起.

在Schoenberg给很多同行的公开信中，他表达了对这一发现的喜爱.此后，H.B.Curry解释了在第二次世界大战期间，他与Schoenberg合作的关于一元B样条的理论的研究[80].我们在本书的最后附上“样条函数之父”Schoenberg的信件，作为多元样条发展历史的记录.

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Let me bring you the good news that B-splines and Polya frequency functions are now being studied in higher dimensions.This was recently done by Wolfgang Dahmen,of Bonn,and Charles Micchelli,of IBM.Let me remind you that Polya frequency functions de Bruijn called them’Polyamials’-were characterized in four different but equivalent ways:I.As limits of B-splines with appropriate knots as their degree tends to inf inity,II.As totally positive functions(frequency functions),III.By their variation diminishing property on convolution,IV.By their Laplace transforms being reciprocals of entire functions of the Laguerre-Polya class.At the same time a B-spline Mn of degree n-1,having n knots,was known to be the orthogonal projection onto the real axis of an n-dimensional simplex S,of volume 1.Now Dahmen and Micchelli are projecting S onto the lower-dim,space Rs,obtaining the s-dimensional B-spline Mn,s.Keeping s f ixed and letting n tend to inf inity,they obtain as limits of M n,s the s-dimensional Polya frequency functions s,and also their Laplace transforms as the reciprocalsof a class Esof entire functions of s variables.This class Es turns out to be identical with the class of“lineal”functions studied by Motzkin and myself in 1952.Lineal functions were an extension of results of Laguerre and Polya,and B-splines Mn,s and Polya frequency functions fs,were not even mentioned in 1952.Now we know where this class Es is coming from.This work extends to Rs the characterizations I and IV.Very likely also II and III will appear by means of restrictions of f s to appropriate lines or planes.I am very happy with these developments.