A new method of analysis of discrete digital experimental data

本课题主要研究和利用我们最近为每个半单李群G引入的三个特殊函数族(称为C-函数、S-函数和E-函数)的性质，其中变量个数n是群的秩数。除了理论物理中大多数特殊函数的共同性质外，还有两个性质使这些函数在现代应用中特别有用。(I)离散化。新函数在实n维欧氏空间中很容易离散化。也就是说，当在有限区域F上积分时，它们的连续正交性扩展到当在任意密度的F中的离散网格上求和时的正交性和G的权重晶格的对称性。因此，数字n维数据/函数可以扩展成一系列新的函数，并且可以计算展开的系数。(Ii)对任何维度和对称性的晶格进行平滑的数字数据内插。德·蒙特利尔。在项目中提出了使用新的特殊功能的一些理论和应用问题。其他课题，多年来一直是我研究的一部分，将在未来五年内研究，包括：(I)利用最近获得的经典李代数的分次知识；(Ii)保持特定等级的李代数的收缩及其表示；(Iii)二维和三维确定性非周期点集(准晶)的研究。

The project is focused on the study and exploitation of properties of three families of special functions (called C-, S-, and E-functions) we have introduced recently for each semisimple Lie group G.  The number of variables n is the rank of the group. Besides the properties which are common to most of special functions of theoretical physics, there are two which make these functions particularly useful in modern applications. (i) Discretization.  The new functions are readily discretized in real n-dimensional Euclidean space. That is, their continuous orthogonality, when integrated over a finite region F, extends to the orthogonality when summed over a discrete grid in F of any density and of the symmetry of the weight lattice of G. Hence digital n-dimensional data/functions can be expanded into series of the new functions, and the coefficients of the expansion can be calculated. (ii) Smooth interpolation of digital data on lattices of any dimension and symmetry.   General commercial exploitations of this property are being patented by Univ. de Montreal. A number of theoretical and applied problems, that will use the new special functions, are presented in the project. Other topics, which have been part of my research for many years and which will be studied during the next five years,  are the following: (i) exploitation of recently acquired knowledge of gradings of classical Lie algebras; (ii) contractions of Lie algebras and their representations with a particular grading preserved; (iii) exploitation of deterministic aperiodic point sets (`quasicrystals') in 2 and 3 dimensions.