  The discrete cosine and sine transforms are generalized to a triangular
fragment of the honeycomb lattice. The honeycomb point sets are constructed by
subtracting the root lattice from the weight lattice points of the
crystallographic root system $A_2$. The two-variable orbit functions of the
Weyl group of $A_2$, discretized simultaneously on the weight and root
lattices, induce a novel parametric family of extended Weyl orbit functions.
The periodicity and von Neumann and Dirichlet boundary properties of the
extended Weyl orbit functions are detailed. Three types of discrete complex
Fourier-Weyl transforms and real-valued Hartley-Weyl transforms are described.
Unitary transform matrices and interpolating behaviour of the discrete
transforms are exemplified. Consequences of the developed discrete transforms
for transversal eigenvibrations of the mechanical graphene model are discussed.
