Discrete cosine and sine transforms generalized to honeycomb lattice

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.

Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

“…x and ε ,s x are listed for the (anti)symmetric cosine and sine sets of points F ,c,± N and F ,s,± N via the weight ε− and ε−functions (28) and ( 29) in Table 2. For the symmetric trigonometric transforms, the number of permutations in the stabilizer Stab S n (y)…”

“…Moreover, the cubature rules [23] of the multivariate Chebyshev polynomials, that are obtained from the (anti)symmetric trigonometric functions [9,10] and associated with the Jacobi polynomials [5,24,25], are further intertwined with the Lie theoretical approach [8,26,27]. The role of the 2D and 3D Fourier-Weyl transforms as tools for solutions of the lattice vibration and electron propagation models [28,29] implies comparable function and direct applicability of the (anti)symmetric trigonometric transforms in solid state physics [30,31] and quantum field theory [32]. The potential diverse applications of both types of discrete transforms also involve image compression [33], laser optics [34], fluid flows [35], magnetostatic modeling [36], and micromagnetic simulations [37].…”

On electron propagation in triangular graphene quantum dots

“…The purpose of this article is to formulate and analytically solve classes of tight-binding models [8,21] that describe an electron propagating in single-layer triangular graphene quantum dots with armchair and zigzag edges [2,16,24,61]. Extending the results for the armchair dots [53] via the honeycomb Weyl orbit functions and the corresponding discrete Fourier-Weyl transforms [26,28], explicit exact forms of the electron wave functions and energy spectra are constructed.…”

“…The point sets describing the atomic positions of the armchair and zigzag triangular graphene dots are advantageously expressed via combinations of A 2 rescaled root and weight lattices [26,28]. Linked to the affine Weyl groups [3,32], discrete Fourier-Weyl transforms utilizing Weyl orbit functions [36,37] have been recently developed on general finite fragments of root and weight lattices [27,30,31,40] as well as subsequently adapted to triangular honeycomb dots [26,28].…”

FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

“…Although Chebyshev polynomials in one variable are ubiquitous in applied mathematics, their multivariate counterparts only recently started to penetrate into applications. Meanwhile, there are applications to the discretization of partial differential equations [31,32,46], cubature formulas [16,23,29] and discrete transforms [2,14,15]. From an algebraic signal processing perspective they are interesting for two reasons.…”