On discretization of tori of compact simple Lie groups. II (Q2901535)

The authors study discrete Fourier-like transforms of data sampled on lattices of any dimension and any symmetry. The problem is to find expansion functions that are complete and orthogonal over finite subsets of the lattices. In the first part of this research, \textit{J. Hrivnák} and \textit{J. Patera} [J. Phys. A, Math. Theor. 42, No. 38, Article ID 385208 (2009; Zbl 1181.65152)] have derived the discrete orthogonality of so-called \(C\)- and \(S\)-functions from the characters of compact simple Lie groups. Now these results are extended to so-called \(S^s\)- and \(S^l\)-functions. These new functions open new possibilities of discrete Fourier-like transforms for the same data. The main result is the discrete orthogonality of the \(S^s\)- and \(S^l\)-functions sampled on finite subsets \(F_M^s\) and \(F_M^l\) of a lattice \(F_M\) in any dimension \(\geq 2\) and of any density controlled by \(M\). Analogous to the discrete Fourier transform, discrete \(S^s\)- and \(S^l\)-transforms are defined.