Three dimensional C-, S- and E-transforms

Details Three dimensional C-, S- and E-transforms Three dimensional C-, S- and E-transforms

2008-05-23

arxiv.org/abs/0805.3731v1

Physics

Mathematical Physics

24 pages, 13 figures

Scientific paper

10.1088/1751-8113/41/47/475205

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-, $S$-, and $E$-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region $F$ of the 3-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid $F_M\subset F$ (discrete orthogonality). The positive integer $M$ sets up the density of the lattice containing $F_M$. The expansion of functions given either on $F$ or on $F_M$ is the paper's main focus.

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