Abstract

An efficient construction is to be described of lattice points FM of any density and any admissible symmetry in a finite region F of a real n-dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n. The density of FM is fixed by our choice of a positive integer M, where 1 ≤ M < ∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM.

Introduction

The goal of this chapter is to provide all of the details necessary for construction of an n-dimensional lattice LM of any symmetry and density in the real Euclidean space ℝn. The motivation for such a construction might be the need to process digital data on LM. This typically requires a system of orthogonal functions on a finite fragment FM ⊂ LM. Such functions are available although their description is outside of the scope of this chapter. Some of the functions are shown here. But for their properties one needs to go to the references provided.

The starting point is a compact simple Lie group G of rank n, or equivalently, the corresponding simple Lie algebra g. Symmetry of its weight lattice P(g) is the symmetry of the lattice LM we construct. Density of LM is determined by our choice of natural number M, that is LM = P(g)/M.