多元指数磨光算子的构造和相关偏微分方程基本解与磨光核的升维解法

CONSTRUCTION OF MULTIVARIATE EXPONENTIAL SMOOTHING OPERATORS AND RAISING DIMENSION TECHNIQUE FOR FINDING THE FUNDAMENTAL SOLUTION OF THE ASSOCIATED PDE AND SMOOTHING KERNEL

本文目的在于回答:δ分布的多元指数磨光函数,即磨光核函数的解析表示问题.从我们给出的多元指数磨光算子的定义出发,将磨光核函数的表示,归结为先求相应偏微分方程的基本解,再对它的广义差分.然后用我们提出的"升维方法",彻底解决了基本解的解析表达问题.从而也就回答了磨光核函数的解析表示.磨光核函数的支集既可以是高维立方体,也可以是高维单纯形.因此,多元指数箱(E-Box)和单纯形(E-Simplex)样条的表示,皆能用我们的统一方法解决.

The aim of the paper is to give the analytic expression of 5-smoothing, i.e.smoothing kernel, by multivariate exponential smoothing operators（MESO）. From the definition of the MESO by us, the construction of smoothing kernel is led to find the fundamental solution of the associated PDE, and its generalized difference. The analytic expression of the later is obtained by our ＂raising dimension technique＂. The support of the smoothing kernel may be high dimensional parallelohedron, or simplex as well. Thus, as a consequence, the problems of analytic expression of exponential box, and simplex splines are all resolved by our united technique.