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。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。(我)离散化。新函数在实际n维欧几里德空间中易于离散化。也就是说，它们的连续正交性，当在有限区域F上积分时，扩展到在F中任意密度的离散网格和g的权格对称上求和时的正交性。因此，数字n维数据/函数可以展开成新函数的系列，并且可以计算展开的系数。（ii）在任何尺寸和对称的格上对数字数据进行平滑插值。蒙特利尔大学正在为这一财产的一般商业开发申请专利。在本项目中提出了一些将使用这些新的特殊函数的理论和应用问题。其他课题，已成为我多年研究的一部分，并将在未来五年内进行研究，如下：(i)利用最近获得的经典李代数分级知识；（ii）李代数的压缩及其保留特定等级的表示；（iii）利用2维和3维的确定性非周期点集（“准晶体”）。

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.