Discrete cosine transforms generalized to honeycomb lattice

# 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 two-variable orbit functions of the Weyl group , discretized simultaneously on the weight and root lattices, induce the family of the 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.

Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, CZ–115 19 Prague, Czech Republic

E-mail: jiri.hrivnak@fjfi.cvut.cz, lenka.motlochova@fjfi.cvut.cz

Keywords: Weyl orbit functions, honeycomb lattice, discrete cosine transform, Hartley transform

## 1. Introduction

The purpose of this article is to generalize the discrete cosine and sine transforms [2] to a finite fragment of the honeycomb lattice. Two-variable (anti)symmetric complex-valued Weyl and the real-valued Hartley-Weyl orbit functions [19, 20, 10, 11] are modified by six extension coefficients sequences and the resulting family of the extended functions contains the set of the discretely orthogonal honeycomb orbit functions. The triangularly shaped fragment of the honeycomb lattice forms the set of nodes of the honeycomb Fourier-Weyl and Hartley-Weyl discrete transforms.

The one-variable discrete cosine and sine transforms and their multivariate concatenations constitute the backbone of the digital data processing methods [2]. The periodicity and (anti)symmetry of the cosine and sine functions represent intrinsic symmetry properties inherited from the (anti)symmetrized one-variable exponential functions. These symmetry characteristics, essential for data processing applications, restrict the trigonometric functions to a bounded interval and induce boundary behaviour of these functions at the interval endpoints. The boundary behaviour of the most ubiquitous two-dimensional cosine and sine transforms is directly induced by the boundary features of their one-dimensional versions [2]. Two-variable symmetric and antisymmetric orbit functions of the crystallographic reflection group , confined by their symmetries to their fundamental domain of an equilateral triangle shape, satisfy similar Dirichlet and von Neumann boundary conditions [19, 20]. These fundamental boundary properties are preserved by the novel parametric sets of the extended Weyl and Hartley-Weyl orbit functions. Narrowing the classes of the extended Weyl and Hartley-Weyl orbit functions by three non-linear conditions, continua of parametric sets of the discretely orthogonal honeycomb orbit functions and the corresponding discrete transforms are found.

Discrete transforms of the Weyl orbit functions over finite sets of points and their applications are, of recent, intensively studied [14, 12, 13, 15, 22]. The majority of these methods arise in connection with point sets taken as the fragments of the refined dual-weight lattice. Discrete orthogonality and transforms of the Weyl orbit functions over a fragment of the refined weight lattice are formulated in connection with the conformal field theory in [15]. Discrete orthogonality and transforms of the Hartley-Weyl orbit functions over a fragment of the refined dual weight lattice are obtained in a more general theoretical setting in [10] and explicit formulations of discrete orthogonality and transforms of both Weyl and Hartley-Weyl functions over a fragment of the refined dual root lattice are achieved in [11]. For the crystallographic root system , the dual weight and weight lattices as well as the dual root and root lattices coincide and the weight lattice of , with the root lattice points omitted, constitutes the honeycomb lattice. Thus, recent explicit formulations of the discrete orthogonality relations of the Weyl and Hartley-Weyl orbit functions over both root and weight lattices of permit construction of the discrete honeycomb transforms.

The potential of the developed Fourier-like discrete honeycomb transforms lies in the data processing methods on the triangular fragment of the honeycomb lattice as well as in theoretical description of the properties of the graphene material [3]. Both transversal and longitudinal vibration modes of the graphene are regularly studied assuming Born–von Kármán periodic boundary condidions [4, 6, 7, 26]. Special cases of the honeycomb orbit functions potentially represent vibration modes that satisfy additional Dirichlet and von Neumann conditions on the triangular boundary. Since spectral analysis provided by the developed discrete honeycomb transforms enjoys similar boundary properties as the 2D discrete cosine and sine transforms, the output analysis of the graphene-based sensors [27, 17] by the honeycomb transforms offers similar data processing potential. The honeycomb discrete transforms provide novel possibilities for study of distribution of spectral coefficients [18], image watermarking [9], encryption [23], and compression [8] techniques.

The paper is organized as follows. In Section 2, a self-standing review of inherent lattices and extensions of the corresponding crystallographic reflection group is included. In Section 3, the fundamental finite fragments of the honeycomb lattice and the weight lattice are described. In Section 4, two types of discrete orthogonality relations of the Weyl and Hartley-Weyl orbit functions are recalled. In Section 5, definitions of the extended and honeycomb orbit functions together with the proof of their discrete orthogonality are detailed. In Section 6, three types of the honeycomb orbit functions are exemplified and depicted. In Section 7, the four types of the discrete honeycomb lattice transforms and unitary matrices of the normalized transforms are formulated. Comments and follow-up questions are covered in the last section.

## 2. Infinite extensions of Weyl group

### 2.1. Root and weight lattices

Notation, terminology, and pertinent facts about the simple Lie algebra stem from the theory of Lie algebras and crystallographic root systems [16, 1]. The basis of the dimensional Euclidean space , with its scalar product denoted by , comprises vectors characterized by their lenghts and relative angle as

 ⟨α1,α1⟩=⟨α2,α2⟩=2,⟨α1,α2⟩=−1.

In the context of simple Lie algebras and root systems, the vectors and form the simple roots of . In addition to the basis of the simple roots, it is convenient to introduce basis of of vectors and , called the fundamental weights, satisfying

 ⟨ωi,αj⟩=δij,i,j∈{1,2}. (1)

The vectors and , written in the basis, are given as

 ω1=23α1+13α2,ω2=13α1+23α2,

and the inverse transform is of the form

 α1=2ω1−ω2,α2=−ω1+2ω2.

The lenghts and relative angle of the vectors and are determined by

 ⟨ω1,ω1⟩=⟨ω2,ω2⟩=23,⟨ω1,ω2⟩=13, (2)

and the scalar product of two vectors in basis and is derived as

 ⟨x,y⟩=13(2x1y1+x1y2+x2y1+2x2y2). (3)

The lattice , referred to as the root lattice, comprises all integer linear combinations of the basis,

 Q=Zα1+Zα2.

The lattice , called the weight lattice, consists of all integer linear combinations of the basis,

 P=Zω1+Zω2.

The root lattice is disjointly decomposed into three shifted copies of the root lattice as

 P=Q∪{ω1+Q}∪{ω2+Q}. (4)

The reflections , , which fix the hyperplanes orthogonal to and pass through the origin are linear maps expressed for any as

 rix=x−⟨αi,x⟩αi. (5)

The associated Weyl group of is a finite group generated by the reflections and . The Weyl group orbit , constituted by images of the point , is given in basis as

 Wx={(x1,x2),(−x1,x1+x2),(−x1−x2,x1),(−x2,−x1),(x2,−x1−x2),(x1+x2,−x2)}. (6)

The root lattice and the weight lattice are Weyl group invariant,

 WP=P,WQ=Q.

### 2.2. Affine Weyl group

The affine Weyl group of extends the Weyl group by shifts by vectors from the root lattice ,

 WaffQ=Q⋊W.

Any element acts on any as

 T(q)w⋅x=wx+q.

The fundamental domain of the action of on , which consists of exactly one point from each orbit, is a triangle with vertices ,

 FQ ={x1ω1+x2ω2|x1,x2≥0,x1+x2≤1}. (7)

For any , the point sets and are defined as finite fragments of the refined lattices and contained in ,

 FP,M =1MP∩FQ, (8) FQ,M =1MQ∩FQ. (9)

The point sets and are of the following explicit form,

 FP,M ={s1Mω1+s2Mω2|s0,s1,s2∈Z≥0,s0+s1+s2=M}, (10) FQ,M ={s1Mω1+s2Mω2|s0,s1,s2∈Z≥0,s0+s1+s2=M,s1+2s2=0mod3}.

Note that the points from are described by the coordinates from (10) as

 s=[s0,s1,s2]∈FP,M (11)

and the set contains only such points from , which satisfy the additional condition . The number of points in the point sets and are calculated in [14, 11] as

 |FP,M| =12(M2+3M+2), (12) ∣∣FQ,M∣∣ ={16(M2+3M+6)M=0mod3,16(M2+3M+2)otherwise. (13)

Interiors of the point sets and contain the grid points from the interior of ,

 ˜FP,M =1MP∩int(FQ), (14) ˜FQ,M =1MQ∩int(FQ). (15)

The explicit forms of the point sets interiors and are the following,

 ˜FP,M ={s1Mω1+s2Mω2|s0,s1,s2∈N,s0+s1+s2=M}, (16) ˜FQ,M Missing \left or extra \right

and the counting formulas from [14, 11] calculate the number of points for as

 ∣∣˜FP,M∣∣ =12(M2−3M+2), (17) ∣∣˜FQ,M∣∣ ={16(M2−3M+6)M=0mod3,16(M2−3M+2)otherwise. (18)

The point sets and are for depicted in Figure 1.

A discrete function is defined by its values on coordinates (11) of in Table 1.

### 2.3. Extended affine Weyl group

The extended affine Weyl group of extends the Weyl group by shifts by vectors from the weight lattice ,

 WaffP=P⋊W.

Any element acts on any as

 T(p)w⋅x=wx+p.

For any , the abelian group ,

 ΓM={γ0,γ1,γ2},

is a finite cyclic subgroup of with its three elements given explicitly by

 γ0=T(0)1,γ1=T(Mω1)r1r2,γ2=T(Mω2)(r1r2)2. (19)

The fundamental domain of the action of on , which consists of exactly one point from each orbit, is a subset of in the form of a kite given by

 FP= {x1ω1+x2ω2∈FQ|(2x1+x2<1,x1+2x2<1)∨(2x1+x2=1,x1≥x2)},

For any , the weight sets and are defined as finite fragments of the lattice contained in the magnified fundamental domains and , respectively,

 ΛQ,M=P∩MFQ, (20) ΛP,M=P∩MFP. (21)

The weight set is of the following explicit form,

 ΛQ,M= {λ1ω1+λ2ω2|λ0,λ1,λ2∈Z≥0,λ0+λ1+λ2=M}

and thus, the points from are described as

 λ=[λ0,λ1,λ2]∈ΛQ,M. (22)

The weight set is of the explicit form,

 ΛP,M= {[λ0,λ1,λ2]∈ΛQ,M|(λ0>λ1,λ0>λ2)∨(λ0=λ1≥λ2)}. (23)

The numbers of points in the weight sets and are proven in [14, 11] to coincide with the number of points in and , respectively,

 Missing or unrecognized delimiter for \right (24)

The action of the group on a weight coincides with a cyclic permutation of the coordinates ,

 γ0[λ0,λ1,λ2]=[λ0,λ1,λ2],γ1[λ0,λ1,λ2]=[λ2,λ0,λ1],γ2[λ0,λ1,λ2]=[λ1,λ2,λ0], (25)

and the weight set is tiled by the images of under the action of ,

 ΛQ,M=ΓMΛP,M. (26)

The subset contains only the points stabilized by the entire ,

 ΛfixM={λ∈ΛP,M|ΓMλ=λ}.

Note that there exists at most one point from , which is fixed by . From relation (25), such a point satisfies and, consequently, the set is empty if is not divisible by , otherwise it has exactly one point,

 ∣∣ΛfixM∣∣={1M=0mod3,0otherwise. (27)

Interiors and of the weight sets and contain only points belonging to the interior of the magnified fundamental domain ,

 ˜ΛQ,M =P∩int(MFQ), (28) ˜ΛP,M =P∩MFP∩int(MFQ). (29)

The explicit forms of the interiors of the weight sets are given as

 ˜ΛQ,M= {[λ0,λ1,λ2]∈ΛQ,M|λ0,λ1,λ2∈N}, ˜ΛP,M= {[λ0,λ1,λ2]∈˜ΛQ,M|(λ0>λ1,λ0>λ2)∨(λ0=λ1≥λ2)}. (30)

The numbers of weights in the interior weight sets and are proven in [14, 11] to coincide with the number of points in interiors and , respectively,

 ∣∣˜ΛQ,M∣∣=∣∣˜FP,M∣∣,∣∣˜ΛP,M∣∣=∣∣˜FQ,M∣∣. (31)

The weight sets and are depicted in Figure 1.

A discrete function is defined by its values on coordinates (22) of in Table 2. The function depends only on the number of zero-valued coordinates and thus, is invariant under permutations of ,

 hM(γλ)=hM(λ),γ∈ΓM. (32)

## 3. Point and weight sets

### 3.1. Point sets Hm and ˜HM

For any , the point set is defined as a finite fragment of the honeycomb lattice contained in ,

 HM=1M(P∖Q)∩FQ.

Equivalently, the honeycomb lattice fragment is obtained from the point set by omitting the points of ,

 HM=FP,M∖FQ,M. (33)

Introducing the following two point sets,

 H(1)M= 1M(ω1+Q)∩FQ, H(2)M= 1M(ω2+Q)∩FQ,

their disjoint union coincides due to (4) with the point set ,

 HM=H(1)M∪H(2)M. (34)

The explicit description of is directly derived from (10) and (33),

 HM={s1Mω1+s2Mω2|s0,s1,s2∈Z≥0,s0+s1+s2=M,s1+2s2≠0mod3}.
###### Proposition 3.1.

The number of points in the point set is given by

 |HM|={13(M2+3M)M=0mod3,13(M2+3M+2)otherwise. (35)
###### Proof.

Relation (33) implies that the number of points in is calculated as

 |HM|=|FP,M|−∣∣FQ,M∣∣,

and equation (35) follows from counting formulas (12) and (13). ∎

The interior contains only the points of belonging to the interior of ,

 ˜HM=1M(P∖Q)∩int(FQ)

and thus, it is formed by points from which are not in ,

 ˜HM=˜FP,M∖˜FQ,M. (36)

The explicit form of is derived from (16) and (36),

 ˜HM={s1Mω1+s2Mω2|s0,s1,s2∈N,s0+s1+s2=M,s1+2s2≠0mod3}

and counting formulas (17), (18) and (36) yield the following proposition.

###### Proposition 3.2.

The number of points in the point set for is given by

 ∣∣˜HM∣∣={13(M2−3M)M=0mod3,13(M2−3M+2)otherwise. (37)

The point sets and are for depicted in Figure 2.

### 3.2. Weight sets Lm and ˜LM

For any , the weight set contains the points , which are not stabilized by  ,

 LM=ΛP,M∖ΛfixM. (38)

Relations (23) and (38) yields the explicit form of ,

 LM= {[λ0,λ1,λ2]∈ΛQ,M|(λ0>λ1,λ0>λ2)∨(λ0=λ1>λ2)}.
###### Proposition 3.3.

The number of points in the weight set is given by

 |LM|=12|HM|. (39)
###### Proof.

Formula (38) implies for the number of points that

 |LM|=|ΛP,M|−∣∣ΛfixM∣∣.

Using formulas (13), (24) and (27), the number of points in is equal to

 |LM|={16(M2+3M)M=0mod3,16(M2+3M+2)otherwise. (40)

Direct comparison of counting relations (40) and (35) guarantees (39). ∎

The interior contains the points , which are not stabilized by ,

 ˜LM=˜ΛP,M∖ΛfixM. (41)

The explicit form of is derived from (30) and (41),

 ˜LM= {[λ0,λ1,λ2]∈˜ΛQ,M|(λ0>λ1,λ0>λ2)∨(λ0=λ1>λ2)}

and formulas (18), (27), (31) and (37) yield the following proposition.

###### Proposition 3.4.

The number of points in the grid for is given by

 ∣∣˜LM∣∣=12∣∣˜HM∣∣.

The weight sets and are for depicted in Figure 3.

## 4. Weyl orbit functions

### 4.1. C− and S−functions

Two families of complex-valued smooth functions of variable , labelled by , are defined via one-variable exponential functions as

 Φb(x) =∑w∈We2πi⟨wb,x⟩, (42) φb(x) =∑w∈Wdet(w)e2πi⟨wb,x⟩. (43)

Properties of the Weyl orbit functions have been extensively studied in several articles [19, 20]. The functions (42) and (43) are called and functions, respectively. Explicit formulas for and functions, with a weight and a point in basis, are derived by employing scalar product formula (3) and Weyl orbit expression (6),

 Φb(x)= e23πi((2b1+b2)x1+(b1+2b2)x2)+e23πi(−b1+b2)x1+(b1+2b2)x2)+e23πi((−b1−2b2)x1+(b1−b2)x2) +e23πi((−b1−2b2)x1+(−2b1−b2)x2)+e23πi((−b1+b2)x1+(−2b1−b2)x2)+e23πi((2b1+b2)x1+(b1−b2)x2), (44) φb(x)= e23πi((2b1+b2)x1+(b1+2b2)x2)−e23πi(−b1+b2)x1+(b1+2b2)x2)+e23πi((−b1−2b2)x1+(b1−b2)x2) −e23πi((−b1−2b2)x1+(−2b1−b2)x2)+e23πi((−b1+b2)x1+(−2b1−b2)x2)−e23πi((2b1+b2)x1+(b1−b2)x2).

Recall from [19, 20] that and functions are (anti)symmetric with respect to the Weyl group, i.e. for any it holds that

 Φb(wx) =Φb(x), φb(wx)=det(w)φb(x), (45)

Furthermore, both families are invariant with respect to translations by any ,

 Φb(x+q)=Φb(x),φb(x+q)=φb(x). (46)

Relations (45) and (46) imply that the Weyl orbit functions are (anti)symmetric with respect to the affine Weyl group and, thus, they are restricted only to the fundamental domain (7) of the affine Weyl group. Moreover, the functions vanish on the boundary of and the normal derivative of the functions to the boundary of is zero.

Denoting the Hartley kernel function by

 casα=cosα+sinα,α∈R, (47)

the Weyl orbit functions are modified [11, 10] as

 ζ1b(x) =∑w∈Wcas(2π⟨wb,x⟩), (48) ζeb(x) =∑w∈Wdet(w)cas(2π⟨wb,x⟩). (49)

The (anti)symmetry relation (45) and shift invariance (46) are preserved by the Hartley functions,

 ζ1b(wx) =ζ1b(x),ζeb(wx)=det(w)ζeb(x), (50) ζ1b(x+q) =ζ1b(x),ζeb(x+q)=ζeb(x). (51)

Therefore, the Hartley functions vanish on the boundary of and the normal derivative of the Hartley functions to the boundary of is also zero.

### 4.2. Discrete orthogonality on Fp,m and ˜FP,M

Using coefficients from Table 1, a scalar product of two functions on the refined fragment of the weight lattice (8) is defined as

 ⟨f,g⟩FP,M=∑s∈FP,Mε(s)f(s)¯¯¯¯¯¯¯¯¯g(s). (52)

Discrete orthogonality relations of the functions (42) and Hartley functions (48), labelled by the weights from the weight set (20) and with respect to the scalar product (52), are derived in [14, 10]. The discrete orthogonality relations are for any of the form

 ⟨Φλ,Φλ′⟩FP,M =18M2hM(λ)δλλ′, (53)