A new method of analysis of discrete digital experimental data

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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.