  Four types of discrete transforms of Weyl orbit functions on the finite point
sets are developed. The point sets are formed by intersections of the dual-root
lattices with the fundamental domains of the affine Weyl groups. The finite
sets of weights, labelling the orbit functions, obey symmetries of the dual
extended affine Weyl groups. Fundamental domains of the dual extended affine
Weyl groups are detailed in full generality. Identical cardinality of the point
and weight sets is proved and explicit counting formulas for these
cardinalities are derived. Discrete orthogonality of complex-valued Weyl and
real-valued Hartley orbit functions over the point sets is established and the
corresponding discrete Fourier-Weyl and Hartley-Weyl transforms are formulated.
