# Discrete cosine and sine transforms generalized to honeycomb lattice

## TL;DR

This paper extends discrete cosine and sine transforms to honeycomb lattice structures using Weyl orbit functions, enabling new Fourier-like transforms with applications to mechanical graphene models.

## Contribution

It introduces a novel family of extended Weyl orbit functions and associated discrete transforms tailored for honeycomb lattice geometries.

## Key findings

- Development of three types of discrete Fourier-Weyl transforms.
- Analysis of boundary properties and periodicity of the transforms.
- Application insights for eigenvibrations in mechanical graphene models.

## Abstract

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.