A new method of analysis of discrete digital experimental data

该项目的重点是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-，S-和E-函数）的性质。 变量的个数n是群的秩。除了理论物理中大多数特殊函数所共有的性质外，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。 新函数很容易在真实的n维欧几里得空间中离散化。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F中的任意密度和G的权格的对称性的离散网格上求和时的正交性。因此，数字n维数据/函数可以扩展成一系列新的函数，并且可以计算扩展的系数。(ii)在任意维度和对称的网格上平滑插值数字数据。 蒙特利尔大学正在为这一财产的一般商业开发申请专利。该项目提出了一些将使用新的特殊函数的理论和应用问题。其他议题，这一直是我的研究多年的一部分，并将在未来五年的研究，如下：（一）利用最近获得的知识分级的经典李代数;（二）收缩的李代数及其表示与一个特定的分级保存;（三）利用确定性非周期点集（“准周期”）在2和3维。

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.