CUBO A Mathematical Journal

Vol.11, No¯ 01, (55–71). March 2009

Discrete Clifford analysis: an overview

Fred Brackx1, Hennie De Schepper,

Frank Sommen and Liesbet Van de Voorde

Clifford Research Group, Department of Mathematical Analysis,

Faculty of Engineering, Ghent University,

Galglaan 2, 9000 Gent, Belgium.

email: Freddy.Brackx@UGent.be

ABSTRACT

We give an account of our current research results in the development of a higher di-

mensional discrete function theory in a Clifford algebra context. On the simplest of

all graphs, the rectangular Zm grid, the concept of a discrete monogenic function is

introduced. To this end new Clifford bases, involving so–called forward and backward

basis vectors and introduced by means of their underlying metric, are controlling the

support of the involved operators. As our discrete Dirac operator is seen to square up

to a mixed discrete Laplacian, the resulting function theory may be interpreted as a

refinement of discrete harmonic analysis. After a proper definition of some topological

concepts, function theoretic results amongst which Cauchy’s theorem and a Cauchy in-

tegral formula are obtained. Finally a first attempt is made at creating a general model

for the Clifford bases used, involving geometrically interpretable curvature vectors.

RESUMEN

Nosotros damos un relato de los resultados de investigación actual en el desarrollo de

la teoría de funciones discretas de dimensión grande en un álgebra de Clifford. Sobre

el mas simple de todos los gráficos, la red de rectangulos Zm, el concepto de fun-

ción monogénica discreta es presentado. Con esta finalidad nuevas bases de Clifford,

envolviendo las bases de vectores llamadas forward and backward, son introducidas

mediante su métrica fundamental, estas controlan el soporte de los operadores en-

vueltos. Como nuestro operador de Dirac discreto puede ser visto como un operador

1Corresponding author: Freddy.Brackx@UGent.be



56 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

Laplaciano discreto mixto, la teoría de funciones resultante puede ser interpretada

como refinamiento de análisis armónico discreto. Después de definir algunos conceptos

topológicos, resultados de teoría de funciones entre los cuales el Teorema de Cauchy

y la fórmula de Cauchy integral son obtenidos. Finalmente, una primera tentativa es

hacer uso de un modelo general de bases de Clifford envolviendo vectores de curvatura

geométricamente interpretables.

Key words and phrases: discrete Clifford analysis, discrete function theory, discrete Cauchy

formula.

Math. Subj. Class.: 30G35.

1 Introduction to the Clifford analysis setting

Clifford analysis (see e.g. [3, 4, 14]) is a higher dimensional function theory centred around the

notion of monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac

operator ∂x, defined below. It is a popular viewpoint to consider this function theory both as a

higher dimensional analogue of the theory of holomorphic functions in the complex plane and as a

refinement of classical harmonic analysis. In order to clarify these statements, let us introduce the

underlying framework.

To this end, let R0,m be endowed with a non–degenerate quadratic form of signature (0,m), let

(e1, . . . ,em) be an orthonormal basis for R
0,m and let R0,m be the real Clifford algebra constructed

over R0,m, see e.g. [22]. The non–commutative multiplication in R0,m is governed by

ejek + ekej = −2δjk, j,k = 1, . . . ,m (1)

A basis for R0,m is obtained by considering for each set A = {j1, . . . ,jh} ⊂ {1, . . . ,m} the ele-

ment eA = ej1 . . .ejh , with 1 ≤ j1 < j2 < ... < jh ≤ m. For the empty set ∅ one puts e∅ = 1,

the identity element. Any Clifford number a in R0,m may thus be written as a =
∑
A
eAaA, aA ∈ R.

When allowing for complex constants, the same set of generators (e1, . . . ,em), still satisfying

the anti–commutation rules (1), also produces the complex Clifford algebra Cm, as well as all real

Clifford algebras Rp,q of any signature (p + q = m).

The Euclidean space R0,m is embedded in R0,m by identifying (x1, . . . ,xm) with the Clifford

vector

x =

m∑

j=1

ejxj



CUBO
11, 1 (2009)

Discrete Clifford analysis: an overview 57

The multiplication of two vectors x and y is given by xy = x • y + x ∧ y with

x • y = −

m∑

j=1

xjyj =
1

2
(xy + yx)

x ∧ y =
∑

i<j

eij(xiyj − xjyi) =
1

2
(xy − yx)

being the scalar valued dot product (equalling the Euclidean inner product up to a minus sign)

and the bivector valued wedge product, respectively. Note that the square of a vector x is scalar

valued and equals the norm squared up to a minus sign: x2 = − < x,x > = −|x|2.

Conjugation in R0,m is defined as the anti-involution for which ēj = −ej, j = 1, . . . ,m. In

particular for a vector x we have x̄ = −x.

The Fourier dual of the vector x is the vector valued first order differential operator

∂x =

m∑

j=1

ej∂xj

called Dirac operator. It is precisely this Dirac operator which underlies the notion of monogenicity

of a function, a notion which may be considered as the higher dimensional counterpart of holo-

morphy in the complex plane. A function f defined and differentiable in an open region Ω of Rm

and taking values in R0,m is called left–monogenic in Ω if ∂x[f] = 0. In what follows, we will use

the concept of inner spherical monogenics; these are homogeneous polynomials Pk(x) of degree k

(k ∈ N), which are moreover monogenic, i.e. for which it holds that ∂x[Pk](x) = 0. Since the Dirac

operator factorizes the Laplacian, ∆ = −∂2
x
, monogenicity may also be regarded as a refinement of

harmonicity; in this sense, spherical monogenics can be seen as refinements of spherical harmonics.

The fundamental group leaving the Dirac operator ∂x invariant is the special orthogonal group

SO(m), doubly covered by the Spin(m) group of the Clifford algebra R0,m. For this reason, the

Dirac operator is called a rotation invariant operator. In the present context, we will refer to this

setting as the continuous case, as opposed to the discrete setting treated in this paper.

Recently, several authors have shown interest in finding an appropriate framework for the

development of discrete counterparts of the basic notions and concepts of Clifford analysis, see a.o.

[15, 16, 9, 10, 12]. Some, yet not all, of these contributions are explicitly oriented towards the

numerical treatment of problems from potential theory and boundary value problems, rather than

towards discrete function theoretic results, see also [17, 18]. In this paper, however, we will aban-

don the path of possible applications in order to focus on the fundamental features of a concrete

model for a Clifford algebra framework in which discrete Dirac operators and the corresponding

discrete function theories can be developed, see also [5, 6]. Seen the above mentioned connection

between continuous Clifford analysis and complex analysis in the plane, special attention should



58 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

be paid to the important property of the discrete Dirac operator factorizing a discrete Laplacian.

This was also the case in the study of holomorphic functions on Z2, see e.g. [13, 19, 8] and, more

recently [20, 21].

Discrete mathematics always involve graphs; here, we will only consider the simplest of all

graphs in Euclidean space, namely the one corresponding to the rectangular Zm grid.

2 Definition of a discrete Dirac operator

As announced above, we will consider the natural graph corresponding to the equidistant grid Zm;

thus a Clifford vector x as introduced above will now only show integer co–ordinates. For the

pointwise discretization of the partial derivatives ∂
∂xj

we then introduce the traditional one–sided

forward and backward differences, respectively given by

∆
+

j
[f](x) = f(. . . ,xj + 1, . . .) − f(. . . ,xj, . . .) = f(x + ej) − f(x), j = 1, . . . ,m

∆
−
j

[f](x) = f(. . . ,xj, . . .) − f(. . . ,xj − 1, . . .) = f(x) − f(x − ej), j = 1, . . . ,m

We then first introduce a discrete Laplacian by its usual definition for an arbitrary connected

graph.

Definition 1. Let f be a function defined on the vertices of a connected graph and let x be such

an arbitrary vertex. Then the action of the discrete Laplace operator on f at x is defined by

∆f(x) =
∑

y∼x

(
f(y) − f(x)

)
=

∑

y∼x
f(y) −

(
#Nx

)
f(x)

where the notation y ∼ x means that there is an edge in the graph under consideration which links

the vertex y to x, and where Nx stands for the neighbourhood of x with respect to the graph, i.e.

the set of all points y ∼ x.

In the present case, with respect to the Zm neighbourhood of x, the above definition explicitly

reads

∆
∗
[f](x) =

m∑

j=1

[
∆

+

j
[f](x) − ∆−

j
[f](x)

]
=

m∑

j=1

[f(x + ej) + f(x − ej)] − 2mf(x) (2)

where we have denoted the corresponding discrete Laplacian by ∆∗; it is usually called the star

Laplacian and involves the values of the considered function at the midpoints of the faces of the

unit cube centred at x. Clearly, with respect to the same grid, but changing the graph, other

discrete Laplacians may be defined, involving e.g. the function values at the vertices of the cube

(the cross Laplacian), or at the midpoints of the ”edges”.



CUBO
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Discrete Clifford analysis: an overview 59

For now, we restrict ourselves to the star Laplacian (2); note that it can also be written as

∆
∗
[f](x) =

m∑

j=1

∆
+

j
∆

−
j

[f](x) =

m∑

j=1

∆
−
j

∆
+

j
[f](x)

When passing to the Dirac operator, we cannot simply combine each discretized partial deriva-

tive, be it forward or backward, with the corresponding basis vector ej, j = 1, . . . ,m, since such

attempts do not serve our aim at developing a discrete function theory in which the notion of dis-

crete monogenicity implies discrete harmonicity, as has been shown in [5]. Instead, an alternative

approach is followed, in which the basis vectors will carry an orientation, just like the forward and

backward differences do. To this end, we need to embed the Clifford algebra R0,m into a bigger

one, with an underlying vector space of the double dimension, e.g. C2m, where we consider 2m

vectors e+
j

and e−
j

, j = 1, . . . ,m, satisfying the following anti–commutator relations:

e
+

j
e
+

k
+ e

+

k
e
+

j
= −2g+

jk
, e

−
j
e
−
k

+ e
−
k
e
−
j

= −2g−
jk
, e

+

j
e
−
k

+ e
−
k
e
+

j
= −2Mjk

where the symmetric tensors (g+
jk

), (g−
jk

) and the general tensor (Mjk) determine the corresponding

metric, see also [12]. Three subsequent assumptions on this metric will now significantly reduce

the degrees of freedom in the choice of the metric scalars.

Assumption 1. The forward and the backward basis vector in each particular cartesian direction

add up to the traditional basis vector in that direction, i.e. e+
j

+ e
−
j

= ej, j = 1, . . . ,m.

Assumption 2. There are no preferential cartesian directions, or: all cartesian directions play

the same role in the metric. This assumption will be referred to as the principle of dimensional

democracy and may be seen as a kind of rotational invariance.

Assumption 3. The positive and negative orientations of any cartesian direction play an equiva-

lent role. This assumption may be interpreted as a kind of reflection invariance.

On the basis of the second and third assumptions, one may put g+
11

= g
+

22
= . . . = g+

mm
=

g
−
11

= g
−
22

= . . . = g−
mm

= λ, where g±
jj

= −(e±
j

)
2, j = 1, . . . ,m, and M11 = M22 = . . . = Mmm = µ,

where 2Mjj = −(e
+

j
e
−
j

+ e
−
j
e
+

j
), j = 1, . . . ,m. Furthermore, also g±

jk
and Mjk, for j 6= k, should

be independent of their subscripts, whence we put g±
jk

= g and Mjk = Mkj = M, j,k = 1, . . . ,m,

j 6= k. The first assumption, combined with the traditional Clifford multiplication rules, then leads

to the additional conditions λ + µ = 1
2

and g + M = 0. Summarizing, the forward and backward

basis vectors e+
j

and e−
j

, j = 1, . . . ,m, will submit to the following multiplication rules:

• e+
j
e
+

k
+ e

+

k
e
+

j
= e

−
j
e
−
k

+ e
−
k
e
−
j

= −2g, j 6= k

• e+
j
e
−
k

+ e
−
k
e
+

j
= 2g, j 6= k

• (e+
j

)
2

= (e
−
j

)
2

= −λ, j = 1, . . . ,m

• e+
j
e
−
j

+ e
−
j
e
+

j
= 2λ − 1, j = 1, . . . ,m



60 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

We are now led to the definition of our discrete Dirac operator.

Definition 2. The discrete Dirac operator ∂ is the first order, Clifford vector valued difference

operator given by

∂ = ∂+ + ∂−

where the forward and backward discrete Dirac operators ∂+ and ∂− are respectively given by

∂+ =

m∑

j=1

e
+

j
∆

+

j
and ∂− =

m∑

j=1

e
−
j

∆
−
j

We obtain, using the above multiplication rules, that

∂2 = −λ

m∑

j=1

(∆
+

j
∆

+

j
+ ∆

−
j

∆
−
j

) + (2λ − 1)

m∑

j=1

∆
+

j
∆

−
j

+ g
∑

j 6=k
(2∆

+

j
∆

−
k
− ∆−

j
∆

−
k
− ∆+

j
∆

+

k
)

If we require the support of ∂2 to remain at least in the unit cube centred at x, the isotropy of the

forward and backward basis vectors needs to be imposed, i.e. we have to put λ = (e+
j

)
2

= (e
−
j

)
2

= 0

as in [12], whence in our case it follows in addition that µ = 1
2
, or e+

j
e
−
j

+e
−
j
e
+

j
= −1, j = 1, . . . ,m.

One thus finally arrives at

• e+
j
e
+

k
+ e

+

k
e
+

j
= e

−
j
e
−
k

+ e
−
k
e
−
j

= −2g, j 6= k

• e+
j
e
−
k

+ e
−
k
e
+

j
= 2g, j 6= k

• (e+
j

)
2

= (e
−
j

)
2

= 0, j = 1, . . . ,m

• e+
j
e
−
j

+ e
−
j
e
+

j
= −1, j = 1, . . . ,m

see also [5]. These relations completely determine the metric of the underlying 2m–dimensional

space in terms of one free scalar parameter g, the metric tensor being given by

mjk =





e
+

j
• e+

k
, j,k = 1, . . . ,m

e
+

j
• e−

k
, j = 1, . . . ,m, k = m + 1, . . . , 2m

e
−
j
• e+

k
, j = m + 1, . . . , 2m, k = 1, . . . ,m

e
−
j
• e−

k
, j,k = m + 1, . . . , 2m



CUBO
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Discrete Clifford analysis: an overview 61

or explicitly:

M =




0 −g · · · −g − 1
2

g · · · g

−g 0
. . .

... g − 1
2

. . .
...

...
. . . 0 −g

...
. . . − 1

2
g

−g · · · −g 0 g · · · g − 1
2

− 1
2

g · · · g 0 −g · · · −g

g − 1
2

. . .
... −g 0

. . .
...

...
. . . − 1

2
g

...
. . . 0 −g

g · · · g − 1
2

−g · · · −g 0




Its determinant reads

det M = (−1)m
(1 + 4g)m−1(1 − 4(m − 1)g)

4m

whence it should hold that g 6= − 1
4

and g 6= 1
4(m−1) , since these specific values would induce a

collapse of dimension; for a further discussion of this phenomenon we refer to Section 7. Under

the above conditions, ∂2 takes the form

∂2 = −

m∑

j=1

∆
+

j
∆

−
j

+ g
∑

j 6=k
(∆

+

j
∆

−
k

+ ∆
+

k
∆

−
j
− ∆−

j
∆

−
k
− ∆+

j
∆

+

k
)

= (4(m − 1)g − 1)∆∗ − 2g
∑

j<k

∆̃jk (3)

where ∆∗ is the star Laplacian (2), and

∆̃jk = f(x + ej + ek) + f(x + ej − ek) + f(x − ej + ek) + f(x − ej − ek) − 4f(x), j < k

each ∆̃jk being interpretable as a cross Laplacian on the corresponding (ej,ek) plane, see also

[12]. Note however that the grid points involved in these additional terms do not respect the

neighbourhood Nx of the vertex x in the originally chosen Z
m graph; we will consider in the

next section the particular case where this term disappears. Anyhow, observe that, if (3) is to be

interpreted as a similar result to the continuous factorization ∂2
x

= −∆, then we should in fact

restrict the metric scalar g to the range [0, 1
4(m−1) [.

3 Special case: the star Laplacian factorized

In the special case of the above approach where g = 0, the defining relations for the forward and

backward basis vectors reduce to

• e+
j

+ e
−
j

= ej, j = 1, . . . ,m

• {e+
j
,e

+

k
} = {e−

j
,e

−
k
} = {e+

j
,e

−
k
} = 0, j,k = 1, . . . ,m, j 6= k



62 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

• (e+
j

)
2

= (e
−
j

)
2

= 0, j = 1, . . . ,m

• {e+
j
,e

−
j
} = −1, j = 1, . . . ,m

(with the usual notation {., .} for the anti–commutator). This particular choice for the metric scalar

causes the second term in (3) to drop, whence we are left with a factorization of the star Laplacian,

i.e. ∂2 = −∆∗, the support of the involved operators now staying in the Zm neighbourhood of x. As

has been remarked in [5, 11], there is a well–known model for these particular forward and backward

vectors, namely the so–called Witt basis of the Clifford algebra C2m. In order to understand this

model properly, provide C2m with the structure of a Hermitean space by introducing a so–called

complex structure J on the underlying orthogonal space R0,2m, i.e. J ∈ SO(2m) with J2 = −1.

For details on the construction, we refer to [1, 2]; for our purpose the following observations are

sufficient. Start from the given orthonormal basis (e1, . . . ,em) of R
0,m and complement it with

additional vectors (em+1, . . . ,e2m) yielding an orthonormal basis of R
0,2m, i.e. ejek +ekej = −2δjk,

j,k = 1, . . . , 2m. Without loss of generality, the complex structure J may always be chosen such

that it maps the m–dimensional subspaces spanned by (e1, . . . ,em) and by (em+1, . . . ,e2m) onto

each other. A commonly used choice is J[ej] = −em+j and J[em+j] = ej, j = 1, . . . ,m, but other

choices are possible as well. The Witt basis (fj, f
c

j
)
m

j=1
for the complex Clifford algebra C2m is then

obtained through the action of the projection operators 1
2
(1 ± iJ) on the basis elements ej:

fj =
1

2
(ej + iJ[ej]) =

1

2
(ej − iem+j), j = 1, . . . ,m

fc
j

=
1

2
(ej − iJ[ej]) =

1

2
(ej + iem+j), j = 1, . . . ,m

It holds that fj +f
c

j
= ej, j = 1, . . . ,m and moreover the Witt basis elements satisfy the Grassmann

identities fjfk + fkfj = f
c

j
fc
k

+ fc
k
fc
j

= 0, j,k = 1, . . . ,m, which also implies their isotropy ((fj)
2

=

(fc
j
)
2

= 0, j = 1, . . . ,m), and the duality identities fjf
c

k
+ fc

k
fj = −δjk, j,k = 1, . . . ,m. These

properties exactly coincide with the above conditions on the vectors e+
j

and e−
j

, so that we may

put e+
j

= fj and e
−
j

= fc
j
, j = 1, . . . ,m and we are left with the Witt discrete Dirac operator

∂ = ∂+ + ∂−, with ∂+ =
∑
m

j=1
fj ∆

+

j
and ∂− =

∑
m

j=1
fc
j
∆

+

j
. This setting was already mentioned

in [21], however without any function theoretic aims.

4 Discrete monogenic functions

In order to define discrete monogenicity, one first needs some discrete topology. So, consider a

bounded set B ⊂ Zm and its characteristic function

ψB(x) =

{
1 if x ∈ B

0 if x /∈ B

as well as the discrete operator

∂̌ =

m∑

j=1

e
+

j
∆

−
j

+

m∑

j=1

e
−
j

∆
+

j



CUBO
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Discrete Clifford analysis: an overview 63

The vector valued function

ψB∂̌ =

m∑

j=1

e
+

j
∆

−
j

[ψB] +

m∑

j=1

e
−
j

∆
+

j
[ψB]

is called the oriented boundary of B. Observe that supp(ψB∂̌) contains points which do not belong

to B. In fact, it consists of all vertices the Zm neighbourhood of which contains points of both B

and co(B) ≡ Zm \ B. In addition to this definition of the boundary, one may then also define the

interior of B (respectively the exterior of B) to be the set of all points of B (respectively of co(B))

which do not belong to supp(ψB∂̌). Each bounded set B ⊂ Z
m thus gives rise to a partition of Zm

into its interior, its exterior and the support of its oriented boundary.

The above concepts now allow to give a definition of a discrete monogenic function.

Definition 3. Let B be a bounded set in Zm and let the Clifford algebra valued function f be

defined on B ∪ supp(ψB∂̌). The f is called discrete (left) monogenic in B if and only if it holds

that ∂[f](x) = 0 for all x ∈ B.

Defined in this way, discrete monogenicity constitutes a proper generalization to higher di-

mension of discrete holomorphy in the Isaacs or the Ferrand sense, see [13, 19]. Moreover, it may

be seen as a refinement of discrete harmonicity, since the right hand side of (3) can be interpreted

as a generalized discrete Laplacian, also called mixed Laplacian, see [12], which even coincides with

the star Laplacian when g = 0.

5 Some function theoretic results

We consider Clifford algebra valued functions defined on Zm.

Then, first of all, a discrete version of Leibniz’s rule is obtained by direct calculation. Observe

that, as compared to its continuous counterpart, it contains an extra term, which fortunately will

turn out to become small when considering finer grids.

Lemma 1 (Leibniz’s rule). Let f and g be Clifford algebra valued functions defined on Zm. Then

(i) ∆±
j

[fg] = (∆
±
j
f) g + f (∆

±
j
g) ± (∆±

j
f) (∆

±
j
g);

(ii) if f is scalar–valued, then

[fg] ∂̌ = g (f∂̌) + f (g∂̌) +

m∑

j=1

(
(∆

+

j
f)(∆

+

j
g)e

−
j
− (∆−

j
f)(∆

−
j
g)e

+

j

)
.



64 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

Next, the integral of a discrete function f is quite naturally defined as
∫
f =

∑

x∈Zm
f(x)

where, in order to ensure integrability, integrands are required to have compact supports. The

following results were then directly obtained, see [6].

Lemma 2 (partial integration). Let f and g be Clifford algebra valued functions defined on Zm,

where at least one of both has compact support, then
∫
f∆

±
j

[g] = −

∫
∆

±
j

[f] g

Lemma 3 (Stokes’ theorem). Let f and g be Clifford algebra valued functions defined on Zm,

where at least one of both has compact support, then
∫
f (∂g) = −

∫
(f∂̌) g and

∫
f (∂̌g) = −

∫
(f∂) g

Observe that the domains of integration on both sides of the formulae in the above lemmata

need not to be the same.

On account of Stokes’ theorem, one now easily arrives at a first fundamental result.

Theorem 1 (Cauchy’s theorem). Let f be a Clifford algebra valued function defined on Zm, which

is discrete left monogenic in the bounded set B, then
∫

(ψB ∂̌) f = 0

Corollary 1. If B is a bounded set in Zm, then
∫
ψB ∂̌ = 0

Clearly, for the further development of this function theory, a Cauchy integral formula is

essential. So, assume that E is the fundamental solution of operator ∂̌, i.e.

E(x) ∂̌ = δ(x) =

{
0, x 6= 0

1 x = 0

}
=

m∏

j=1

δ0 xj (4)

and

E(x − y) ∂̌ = δ(x − y) =

{
0, x 6= y

1 x = y

}
=

m∏

j=1

δxj yj (5)

For further use, we then define

GT(x,y) =

m∑

j=1

(
∆

+

j
[ψB(x)]∆

+

j
[E(x − y)] e−

j
− ∆−

j
[ψB(x)]∆

−
j

[E(x − y)] e+
j

)
(6)

The following results were then obtained in [6].



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Discrete Clifford analysis: an overview 65

Theorem 2 (Cauchy–Pompeiu formula). Let B be a bounded set in Zm and let f be a Clifford

algebra valued function defined on B ∪ supp(ψB ∂̌), then for all points y ∈ B it holds that

−f(y) =

∫
ψB(x) E(x − y)∂f(x) +

∫
E(x − y) (ψB∂̌) f(x) +

∫
GT(x,y) f(x)

while for all points y ∈ co(B):

0 =

∫
ψB(x) E(x − y)∂f(x) +

∫
E(x − y) (ψB∂̌) f(x) +

∫
GT(x,y) f(x)

where GT(x,y) is given by (6).

The first and the second term at the right hand side in the above formulae are ’traditional’

terms, representing a volume integral over the bounded set B and a surface integral over the

oriented boundary of B, respectively. On the contrary, the third term is an additional one, arising

due to the grid (and more precisely: it originates from the additional term already arising in

Leibniz’s rule). We call this term the ’grid tension’ term, which explains the notation GT(x,y),

introduced above.

Theorem 3 (Cauchy’s integral formula). Let B be a bounded set in Zm and let the function f be

discrete monogenic on B, then for all points y ∈ B it holds that

−f(y) =

∫
E(x − y) (ψB∂̌) f(x) +

∫
GT(x,y) f(x)

while for all points y ∈ co(B):

0 =

∫
E(x − y) (ψB∂̌) f(x) +

∫
GT(x,y) f(x)

where GT(x,y) is given by (6).

Obviously, in the above results, an essential role is played by the so–called fundamental solution

E(x), defined by (4)–(5). In order to obtain E(x) explicitly, we will pass to frequency space by

means of the discrete–time Fourier transform, defined for a discrete Clifford algebra valued function

f(x) with compact support as follows:

F[f(x)](ξ) =

∫
f(x) exp(−i〈ξ,x〉) =

∑

x∈Zm
exp(−i〈ξ,x〉) f(x), ξ ∈ Zm (7)

and yielding a periodic function of ξ with period (2π)m. Elementary properties of this discrete–time

Fourier transform are listed in the following lemma.

Lemma 4. Let f(x) be a Clifford algebra valued function defined on Zm with compact support and

let its discrete–time Fourier transform be given by (7), then it holds that

• F[f(x ± ej)](ξ) = exp(±iξj) F[f(x)](ξ);



66 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

• F[∆±
j
f(x)](ξ) = ∓(1 − exp(±iξj)) F[f(x)](ξ);

• F[f(x) ∂̌](ξ) = F[f(x)](ξ) G(ξ), where

G(ξ) =

m∑

j=1

[
(1 − exp(−iξj)) e

+

j
+ (exp(iξj) − 1) e

−
j

]
(8)

• F[δ(x)](ξ) = 1.

On account of these calculus rules, it was then obtained in [6] that

Ê(ξ) ≡ F[E(x)](ξ) =
G(ξ)

(G(ξ))2
, wherever G(ξ) 6= 0 (9)

with G(ξ) being given by (8).

In Section 7, G(ξ) and Ê(ξ) are obtained even more explicitly, when passing to a concrete

model for the Clifford forward and backward bases.

6 Discrete monogenic polynomials

Here our aim is to establish a notion of discrete spherical monogenic, i.e. the discrete counterpart of

a monogenic homogeneous polynomial. To this end, one should observe that, for polynomials, it is

not necessary to distinguish between the continuous and the discrete world. Indeed, if a polynomial

is defined in the continuous variable x ∈ Rm, then it is trivially defined on Zm. Conversely, for

each polynomial P(x), there exists a number N such that, if P(x) is defined on a subset A ⊂ Zm,

with |A| = N, then P(x) is well–defined in the whole of Rm. So we are able to use at the same

time derivatives and differences of polynomials.

For further use, we list a few auxiliary results in this respect, see also [7].

Lemma 5. The operators ∆±
j
− ∂xj , j = 1, . . . ,m, turn a homogeneous polynomial of degree k

into a polynomial of degree (k − 2).

Corollary 2. The operator ∂ − ∂x turns a homogeneous polynomial of degree k into a polynomial

of degree (k − 2).

Corollary 3. A homogeneous polynomial of degree k is left monogenic if and only if the discrete

Dirac operator ∂ turns it into a polynomial of degree (k − 2).

Proposition 1. Let Lk(x) be a polynomial of degree k, and let Pk(x) be its homogeneous part of

degree k, i.e. let

Lk(x) = Pk(x) + Rk−1(x)



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Discrete Clifford analysis: an overview 67

the meaning of Rk−1(x) being obvious. If Lk(x) is discrete monogenic, i.e. ∂[Lk](x) = 0, then

Pk(x) is an inner spherical monogenic, i.e. ∂x[Pk](x) = 0.

Corollary 4. A homogeneous discrete monogenic polynomial is automatically an inner spherical

monogenic.

Although, fortunately, the converse is not true, the above corollary nevertheless indicates that

it makes no sense to define an inner spherical discrete monogenic to be a discrete monogenic homo-

geneous polynomial. The question thus raises if an inner spherical monogenic can be completed,

possibly uniquely, to a discrete monogenic polynomial of the same degree. The answer is given in

the proposition below.

Proposition 2. Let Pk(x) be an inner spherical monogenic of degree k. Then there exists a unique

polynomial Rk−2 of degree k − 2, such that

Qk(x) = Pk(x) − xRk−2(x)

is a discrete monogenic polynomial of degree k.

This induces the following fundamental result.

Theorem 4. A discrete monogenic polynomial Lk(x) of degree k may be uniquely decomposed as

Lk(x) = Qk(x) + Lk−1(x)

where Qk(x) is a discrete monogenic polynomial of degree k showing the specific form

Qk(x) = Pk(x) − xRk−2(x)

Pk(x) being an inner spherical monogenic, and where Lk−1(x) is a discrete monogenic polynomial

of degree (k − 1).

The above observations now give rise to the following definition.

Definition 4. A discrete monogenic polynomial Qk(x) of degree k, showing the specific form

Qk(x) = Pk(x) − xRk−2(x)

Pk(x) being an inner spherical monogenic, is called an inner spherical discrete monogenic of degree

k.

By subsequent application of the above theorem, we may now conclude the following.

Corollary 5. For each discrete monogenic polynomial Lk(x) of degree k, there exists a unique set

of inner spherical discrete monogenics (Qj(x))
k

j=0
, such that

Lk(x) = Qk(x) + Qk−1(x) + . . . + Q1(x) + Q0(x)



68 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

7 A model for the forward and backward basis vectors

In Section 2 we have introduced our discrete Dirac operator with respect to the Zm graph, a crucial

role in its definition being played by the so–called forward and backward Clifford basis vectors e+
j

and e−
j

, j = 1, . . . ,m, for which we have already provided a concrete model in the special case

treated in Section 3. In this section, a feasible model is given in the general case where the metric

scalar g does not equal zero, see also [5].

To this end so–called curvature vectors Bj, j = 1, . . . ,m are introduced, by means of which

one puts

e
+

j
=

1

2
(ej + Bj) and e

−
j

=
1

2
(ej − Bj), j = 1, . . . ,m

meanwhile ensuring that e+
j

+ e
−
j

= ej, j = 1, . . . ,m. As these forward and backward Clifford

vectors should satisfy the relations derived in Section 2, it should hold that





B2
j

= +1

{ej,Bj} = 2(ej • Bj) = 0
j = 1, . . . ,m (10)

and that




{ek,Bj} = 2(ek • Bj) = 0

{Bk,Bj} = 2(Bk • Bj) = −8g
j,k = 1, . . . ,m, j 6= k (11)

Note that the second condition in (10) and the first one in (11) together express the orthogonality

of the space spanned by the curvature vectors and the original m-dimensional space with basis

(e1, . . . ,em). As a consequence, the curvature vectors may be written explicitly as

Bj =

m∑

ℓ=1

b
(ℓ)

j
(iem+ℓ) =

m∑

ℓ=1

b
(ℓ)

j
ǫℓ, j = 1, . . . ,m

where ǫ2
ℓ

= (iem+ℓ)
2

= +1, ℓ = 1, . . . ,m and
∑
m

ℓ=1
(b

(ℓ)

j
)
2

= 1, j = 1, . . . ,m. Note that here

the Clifford dot product of any two curvature vectors equals their Euclidean inner product, these

inner products all being equal to the same scalar −4g. (B1, . . . ,Bm) may thus be interpreted as

a set of vectors on the unit sphere Sm−1 of Rm, containing two by two the same fixed angle α,

with cos(α) = −4g. To this end the metric scalar g needs to be restricted to the interval ] − 1
4
, 1

4
],

creating then a kind of ’umbrella’ of vectors, which will open and close according to varying g. In

particular, if g = 0 then α = π
2
, in agreement with the Witt case of Section 3.



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Discrete Clifford analysis: an overview 69

The above relations (10)–(11) are summarized in the metric tensor M̃:

M̃ =




−1 0 · · · · · · 0

0 −1 0
. . .

...

... 0 −1 0
... 0

...
. . . 0 −1 0

0 · · · · · · 0 −1

+1 −4g · · · · · · −4g

−4g +1 −4g
. . .

...

0
... −4g +1 −4g

...
...

. . . −4g +1 −4g

−4g · · · · · · −4g +1




its entries being equal to the Clifford dot products of the vectors (e1, . . . ,em,B1, . . . ,Bm), in this

specific order. Its determinant equalling (−1)m(1 + 4g)m−1(1−4(m−1)g), we are again confronted

with the non–admissible values − 1
4

and 1
4(m−1) for the metric scalar g, already obtained in Section

2. Indeed, in those cases we no longer dispose of a basis for a 2m–dimensional space: instead, for

g = 1
4(m−1) we have that rank (M̃) = 2m− 1, while for g = −

1

4
we have rank (M̃) = m+ 1. We will

further comment on this from a geometrical point of view. To this end, first take g = − 1
4
. Here

Bk•Bj = 〈Bk,Bj〉 = +1, j,k = 1, . . . ,m, whence their contained angle α is zero. So, the ’umbrella’

completely closes, all curvature vectors coincide and the dimension of the space spanned by them

becomes 1, in accordance with the rank of the metric tensor. In the case where g = 1
4(m−1) , the

rank of M̃ shows that the space spanned by the curvature vectors should be (m − 1)–dimensional,

i.e. they should be on the intersection of the unit sphere Sm−1 with a hyperplane in m–dimensional

space. In [5], the contained angle of the vectors in this situation has been explicitly determined

for dimensions m = 3 and m = 4, showing that it indeed corresponds to the given value of g.

Remark 1. It is worth noting that, in this concrete model for the forward and backward Clifford

bases, one has

G(ξ) =

m∑

j=1

[(1 − cos ξj) Bj + i sin ξj ej]

and

(G(ξ))2 = 4

m∑

j=1

sin
2
ξj

2
− 32g

∑

j<k

sin
2
ξj

2
sin

2
ξk

2



70 F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde CUBO
11, 1 (2009)

whence the fundamental solution Ê(ξ) in frequency space, (9), explicitly reads

Ê(ξ) =
1

4

m∑

j=1

[(1 − cos ξj) Bj + i sin ξj ej]

m∑

j=1

sin
2
ξj

2
− 8g

∑

j<k

sin
2
ξj

2
sin

2
ξk

2

This explicit expression also allows to investigate when the denominator of Ê(ξ) will be zero (i.e.,

when G(ξ) = 0), see [6] for the treatment of low dimensional cases.

It is our intention to extend this first model in a forthcoming paper, taking into account

generalized curvature tensors controlling the support of all involved operators.

Acknowledgement

Financial support of the FWO Vlaanderen, by a special grant ("Krediet aan navorsers", no.

31506208) for research on the subject of "Discrete Clifford analysis" is gratefully acknowledged

by Frank Sommen.

Received: June 2008. Revised: July 2008.

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