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Adv. appl. Clifford alg. 17 (2007), 37–58 c 2006 Birkh¨  auser Verlag Basel/Switzerland 0188-7009/010037-22, published online October 15, 2006 DOI 10.1007/s00006-006-0016-5

Advances in Applied Clifford Algebras

Fischer Decomposition for Difference Dirac Operators N. Faustino∗ and U. K¨ahler† Abstract. We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian. Mathematics Subject Classification (2000). 30G35, 30G25, 39A12, 39A70. Keywords. Difference Dirac operator, Difference Euler operator, Fischer decomposition.

1. Introduction Clifford analysis is a powerful tool to solve some kinds of problems related with vector field analysis. A comprehensive description of Clifford function theory was introduced by F. Brackx, R. Delanghe and F. Sommen in [1] and later by R. Delanghe, F. Sommen and V. Sou˘cek in [2]. In [5, 6], K. G¨ urlebeck and W. Spr¨ oßig proposed some strategies to solve boundary value problems based on the study of existence, uniqueness, representation, and regularity of solutions with the help of an operator calculus. In the same books, the authors introduce also the basic ideas to develop a discrete counterpart of the continuous treatment of boundary value problems with the introduction of a discrete operator calculus in order to find a well-adapted numerical approach. An explicit discrete version of the Borel-Pompeiu formula was presented for dimension n = 3. ∗ Supported by Funda¸ ca ˜o para Ciˆ encia e Tecnologia under PhD-grant no. SFRH/BD/17657/ 2004. † Research (partially) supported by Unidade de Investiga¸ ca ˜o Matem´ atica e Aplica¸co ˜es of the University of Aveiro.

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This was further developed in [7, 8], where K. G¨ urlebeck and A. Hommel developed finite difference potential methods in lattice domains based on the concept of discrete fundamental solutions for the difference Dirac operator which generalizes the work developed by Ryabenkij in [10]. A numerical application of this theory was presented recently by N. Faustino, K. G¨ urlebeck, A. Hommel, and U. K¨ ahler in [3] for the incompressible stationary Navier-Stokes equations. In this paper, the authors proposed a scheme which solves efficiently problems in unbounded domains and show the convergence of the numerical scheme for functions with H¨ older regularity which is a better gain compared with the convergence results for classical difference schemes. Moreover, while all these papers claim to be based on discrete function theoretical approaches, from the concepts of the theory of monogenic functions only the Borel-Pompeiu formula and with it Cauchy’s integral formula were obtained. There is no “real” development of a discrete monogenic function theory up to now. This paper is supposed to be a step in this direction. To this end discrete versions of a Fischer decomposition, Euler and Gamma operators are obtained. For the sake of simplicity we consider in the first part only Dirac operators which contain only forward or backward finite differences. Of course, these Dirac operators do not factorize the classic discrete Laplacian. Therefore, we will consider in the last chapter a different definition of a difference Dirac operator in the quaternionic case (c.f. [7]) which do factorizes the discrete Laplacian. Let us emphasize in the end a major obstacle in the discrete case. While in the continuous case there is only one partial derivative for each coordinate xj we have two finite differences in the discrete case. Therefore, we will have not only one Euler or Gamma operator as in the continuous case, but several. Each one will turn out to be connected to one particular Dirac operator.

2. Preliminaries Let e1 , . . . , en be an orthonormal basis of Rn . The Clifford algebra C0,n is the free algebra over Rn generated modulo the relation x2 = −|x|2 e0 , where e0 is the identity of C0,n . For the algebra C0,n we have the anti-commutation relationship ei ej + ej ei = −2δij e0 , where δij is the Kronecker symbol. In the following we will identify the Euclidean 1 space Rn with C0,n , the space of all vectors of C0,n . This means that each element x of Rn may be represented by x=

n  i=1

xi ei .

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Fischer Decomposition for Difference Dirac Operators

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From an analysis viewpoint one extremely crucial property of the algebra C0,n is −x that each non-zero vector x ∈ Rn has a multiplicative inverse given by |x| 2 . Up to a sign this inverse corresponds to the Kelvin inverseof a vector in Euclidean space. Moreover, given a general Clifford number a = A eA aA , A ⊂ {1, . . . , n} we denote by Sc a = a∅ the scalar part and by Vec a = e1 a1 + . . . + en an the vector part. For all what follows let Ω ⊂ Rn be a bounded domain with a sufficiently smooth  boundary Γ = ∂Ω. Then any function f : Ω → C0,n has a representation f = A eA fA with R-valued components fA . We now introduce the Dirac opern ∂ . This operator is a hypercomplex analogue to the complex ator D = i=1 ei ∂x i Cauchy-Riemann operator. In particular we have that D2 = −∆, where ∆ is the Laplacian over Rn . A function f : Ω → C0,n is said to be left-monogenic if it satisfies the equation (Df )(x) = 0 for each x ∈ Ω. A similar definition can be given for right-monogenic functions. Basic properties of the Dirac operator and left-monogenic functions can be found in [1], [2], [6], and [5]. Now, we need some more facts for our discrete setting. To discretize point∂ in the equidistant lattice with mesh width h > wise the partial derivatives ∂x i n 0, Rh = {mh = (m1 h, . . . , mn h) : m ∈ Zn }, we introduce forward/backward differences ∂h±i : ∂h±i u(mh) = ∓

u(mh) − u(mh ± hei ) h

(1)

These forward/backward differences ∂h±i satisfy the following product rules (∂h±i f g)(mh) = (∂h±i f g)(mh) =

f (mh)(∂h±i g)(mh) + (∂h±i f )(mh)g(mh ± hei ),

f (mh ± hei )(∂h±i g)(mh) + (∂h±i f )(mh)g(mh).

(2) (3)

The forward/backward discretizations of the Dirac operator are given by Dh± =

n  i=1

ei ∂h±i .

In this paper we will also use the following multi-index abbreviations: (mh)(α) := (m1 h)α1 (m2 h)α2 . . . (mn h)αn ; α! := α1 !α2 ! . . . αn !; |α| := α1 + α2 + . . . αn ∂h±αi ei for α = (α1 , α2 , . . . , αn ) =

n

i=1

ei αi .

∂h±ei := ∂h±i ; αi  := ∂h±ei ,

(4)

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3. Fischer Decomposition The basic idea of a Fischer decomposition is to decompose any homogeneous polynomial into monogenic homogeneous polynomials of lower degrees. In the classic case such a decomposition is based on the fact that the powers xs are homogeneous ∂xs and that ∂xii = sxs−1 . A first idea would be to consider instead of xs simply the i powers (mh)s , but while these powers are still homogeneous the last condition is not true in the discrete case, unfortunately. Therefore, we will start by introducing discrete homogeneous powers which will play the equivalent role of xs in the discrete case. 3.1. Multi-index factorial powers Starting from the one-dimensional factorial powers (0)

(s)

(mi h)∓ := 1, (mi h)∓ :=

s−1 

(mi h ∓ kh), s ∈ N

(5)

k=0

we introduce the multi-index factorial powers of degree |α| by (α) (mh)∓

=

n 

(α )

(mi h)∓ i .

i=1 (s)

The one-dimensional factorial powers (mi h)∓ have the following properties P1. P2. P3. P4.

(s+1)

(s)

= (mi h ∓ sh)(mi h)∓ ; (mi h)∓ (s) (s−1) ∂h±j (mi h)∓ = s(mi h)∓ δi,j ; (s) (s−1) ∂h∓j (mi h)∓ = s(mi h ∓ h)∓ δi,j ; (s) (mi h)∓ → xsi = (mi h)s for h → 0,

where δi,j denotes the standard Kronecker symbol. As a direct consequence of these properties, we obtain the following lemmas: (α)

Lemma 3.1. The multi-index factorial powers of degree |α|, (mh)∓ , satisfy n  i=1

(α)

(α)

(mi h)∂h±i (mh ∓ hei )∓ = |α|(mh)∓ . (α)

Lemma 3.2. The multi-index factorial powers of degree |α|, (mh)∓ , satisfy (α)

∂h±β (mh)∓ = α!δα,β

for |β| = |α|. (α)

Lemma 3.3. The multi-index factorial powers (mh)∓ of degree |α| approximate the classical multi-index powers x(α) of degree |α| at each point x = mh.

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Let us remark that we have the following relationships between the multiindex factorial powers and the usual powers: i Theorem 3.1. The powers (mi h)αi and (mi h)α ∓ are related by

αi 

i (mi h)α ∓ =

(mi h)αi =

ki =0 αi  ki =0

Skαii (mi h)ki Tkαii (mi h)k∓i ,

Skαii

where are the Stirling numbers of the first kind and Tkαii are the Stirling numbers of the second kind. The sketch of the proof of this theorem can be found, e.g., in [11]. (α)

Theorem 3.2. The multi-index powers (mh)(α) and (mh)∓ are related by (α) (mh)∓

|α| 

=

Kβα (mh)(β) ,

(6)

|β|=0

(mh)(α) =

|α| 

(β)

Lα β (mh)∓ .

(7)

|β|=0

Moreover, |β| 

Kβα =

ln−1



...

ln−1 =0 ln−2 =0

Lα β =

|β| 

l1 =0

ln−1



...

ln−1 =0 ln−2 =0

l2 

l2  l1 =0

αn Slα11 Slα22−l1 . . . S|β|−l , n−1

αn 2 Tlα1 1 Tlα2 −l . . . T|β|−l . 1 n−1

We will just prove identity (6). The proof of identity (7) is analogous to the proof of identity (6). Proof. Using Theorem 3.1 and multiplying the polynomials (m1 h)α1 and (m2 h)α2 , we obtain α 1 +α2 (β1 ) (β2 ) 2 (m1 h)α1 (m2 h)α2 = Kαβ11 ,β ,α2 (m1 h)∓ (m2 h)∓ ,β2 with Kαβ11 ,α = 2



β1 +β2 l1 =0

β1 +β2 =0

 Slα11 Sβα12+β2 −l1 .

Using again Theorem 3.1 and multiplying the polynomials (m1 h)α1 (m2 h)α2 and (m3 h)α3 , we obtain (m1 h)α1 (m2 h)α2 (m3 h)α3 =

α1 +α 2 +α3  β1 +β2 +β3 =0

(β )

(β )

(β )

1 2 3 2 ,β3 Kαβ11 ,β ,α2 ,α3 (m1 h)∓ (m2 h)∓ (m3 h)∓

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 +β2 +β3 l2 α1 α2 α3 ,β2 ,β3 with Kαβ11 ,α = βl21=0 l1 =0 Sl1 Sl2 −l1 Sβ1 +β2 +β3 −l2 . 2 ,α3 Applying this procedure recursively, we obtain (α)

(mh)

=

|α| 

(α)

Kβα (mh)∓

(8)

|β|=0

with Kβα =

|β|

ln−1 =0

ln−1

ln−2 =0

...

 l2

l1 =0

αn Slα11 Slα22−l1 . . . S|β|−l . n−1



For all what follows, let Π± k denote the space of all Clifford-valued polynomials (α) ± of degree k, Pk , generated by the powers (mh)∓ of degree |α| = k, and Π± be the countable union of all Clifford-valued polynomials of degree k ≥ 0. Furthermore, ± ± let M± k = Πk ∩ ker Dh be the space of discrete monogenic polynomials of degree k. Based on Lemma 3.1, 3.2 and 3.3, we will show that it is possible to obtain discrete versions for the Fischer decomposition as well as define discrete versions of the Euler and Gamma operators. 3.2. The main theorem ± For two Clifford-valued polynomials of degree k, Pk± and Q± k ∈ Πk given by  (α) (mh)∓ a± Pk± (mh) = α |α|=k

Q± k (mh)

=



(α)

(mh)∓ b± α

|α|=k

we define the Fischer inner product by [Pk± , Q± k ]h :=



± α!Sc (a± α bα ).

(9)

|α|=k

Denoting by Pk± (Dh± ) the difference operator obtained from the polynomial in powers of mh by replacing mi h by the difference operator ∂h±i (just like in the continuous case, c.f. [2]), we have by Lemma 3.2 the identity

Pk±

± ± ± [Pk± , Q± k ]h := Sc (Pk (Dh )Qk )(0)

± Pk± , Q± k ∈ Πk .

(10)

With other words, we can express the Fischer inner product by applying the difference operator Pk± (Dh± ) to the polynomial Pk± and evaluate the scalar part at the point mh = 0. Moreover, due to Dh± = −Dh± the Fischer inner product has the important property: ± ± ± [(mh)Pk± , Q± (11) k ]h = −[Pk , Dh Qk ]h . This property allows us to prove the following theorem: Theorem 3.3. We have ± ± Π± k = Mk + (mh)Πk−1 .

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± Moreover, the subspaces M± k and (mh)Πk−1 are orthogonal with respect to the Fischer inner product.

Before we prove Theorem 3.3, we will prove the following inclusion property: ± ± ± ± Lemma 3.4. For the set Dh± Π± k := {Dh Pk : Pk ∈ Πk }, we have the inclusion: ± ± ± ± ± Dh± Π± k := {Dh Pk : Pk ∈ Πk } ⊂ Πk−1 .

(α) ∂h±i (mh)∓

=



(α) ± |α|=k (mh)∓ aα (α−e ) αi (mh)∓ i , the identity

Proof. Let Pk± (mh) =

(Dh± Pk± )(mh) =

± ∈ Π± k . Applying Dh , we obtain from

n  

(α−ei )

(mh)∓

αi ei a± α

(12)

i=1 |α|=k

Because αi ei a± α is a Clifford constant we have a linear combination of polynomials of degree |α−ei | = k −1 on the right hand side of (12). Hence, Dh± Pk± ∈ Π± k−1 .   ⊥ ± ± Proof of Theorem 3.3. Because of Π± it is enough k = (mh)Πk−1 + (mh)Πk−1 ⊥  ± ± ± ± to prove that (mh)Πk−1 = Mk−1 . For this we choose Pk−1 ∈ Πk−1 arbitrarily and assume that for some Pk± ∈ Π± k we have ± [(mh)Pk−1 , Pk± ]h = 0. ± ± , Dh± Pk± ]h = 0 for all Pk−1 . As Dh± Pk± ∈ Π± Due to (9) we have [Pk−1 k−1 by ± ± ± ± ± Lemma 3.4, we obtain Dh Pk = 0 or Pk ∈ Mk . This means that ((mh)Πk−1 )⊥ ⊂ ± ± ± ± M± k . Now, let Pk ∈ Mk . Then we have for each Pk−1 ∈ Πk−1 ± [(mh)Pk−1 , Pk± ]h

± = −[Pk−1 , Dh± Pk± ]h

= 0 ±  and, therefore, ((mh)Π± k−1 ) = Mk−1 .



From this theorem we obtain the Fischer decomposition with respect to our difference Dirac operators Dh± . Theorem 3.4 (Fischer decomposition). Let Pk± ∈ Π± k then Pk± (mh) =

k−1  s=0

± (mh)s Mk−s (mh).

(13)

where each Mj± denotes the homogeneous discrete monogenic polynomials of degree j with respect to the Dirac operators Dh± .

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3.3. Difference Euler and Gamma operators Based on Lemma 3.1 we will introduce discrete versions of the Euler and Gamma operators presented in [2]. First of all, we introduce the second order difference operator A± h by A± h

= ∓h

n  i=1

(mi h)∂h±i ∂h∓i .

(14)

Definition 3.1. For a lattice function fh : Rnh → C0,n , we introduce the difference Euler operator Eh± by (Eh± fh )(mh) =

n  i=1

(mi h)(∂h±i fh )(mh ∓ hei )

and the difference Gamma operator Γ± h by  ± (Γ± ej ek (L± h fh )(mh) = − jk fh )(mh) − (Ah fh )(mh), j 0,    (r) (th ∓ h) fh (mh) = hd± f ((th)(mh)) ∓ h h th∈[0,1]h

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By a direct calculation,   (r) d± h (th ∓ h)∓ fh ((th)(mh)) = =

(r−1)

(r)

fh ((th)(mh)) + (th)∓ (d± h fh )((th)(mh))   (r−1) ± rfh ((th)(mh)) + th(dh fh )((th)(mh)) (th ∓ h)∓

r(th ∓ h)∓

On  the other hand, applying the difference version of the chain rule and the relation ni=1 (mi h)∂h±i = Eh± − A± h , we obtain th(d± h fh )((th)(mh))

= =

n 

±i (th)(mi h)(∂th fh )((th)(mh))

i=1 ± (Eth fh )((th)(mh))

− (A± th fh )((th)(mh))

Therefore, fh (mh)

± ± ± = r(Jh,r fh )(mh) + (Eh± Jh,r fh )(mh) − (A± h Jh,r fh )(mh) ± ± = (Rh,r Jh,r fh )(mh).

± ± and Jh,r , we get From the above two identities and the definitions of Rh,r ± ± ± ± Rh,r = I = Rh,r Jh,r . Jh,r



± Now, the construction of the inverse for Vh,r seems to be obvious. But, the obvious choice    (r−1) ± (Wh,r fh )(mh) = hd± f ((th)(mh)) (th) h ∓ h th∈[0,1]± h ± is not an inverse of Vh,r , which can be easily checked in the following way. If we use the same technique as above, we obtain    (r−1) hd± f ((th)(mh)) (th) fh (mh) = h ∓ h th∈[0,1]± h

and by direct calculation   (r) d± h (th)∓ fh ((th)(mh)) (r−1)

(r)

fh (thx) + (th ± h)∓ (d± h fh )((th)(mh))   (r−1) = (th)∓ rfh ((th)(mh)) + (th ± h)(d± h fh )((th)(mh))

= r(th)∓

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On  the other hand, applying the difference version of the chain rule and the n relation i=1 (mi h)∂h±i = Eh± − A± h , we obtain (th ±

h)(d± h fh )((th)(mh))

=

n  i=1

±i ((th)(mi h) ± hmi )(∂th fh )((th)(mh))

± = (Eth fh )((th)(mh)) − (A± th fh )((th)(mh)) n  ±i mi h(∂th fh )((th)(mh)), ±h i=1

but ±h

n  i=1

±i (mi h)(∂th fh )((th)(mh))

=

±h =

n 

±i (∂th fh )((th)(mh))

i=1 ± (Bth fh )((th)(mh)).

3.4. Difference operator calculus Now we will establish some properties for our difference operators introduced in Section 3.3. Using the difference properties (α)

(α)

(mi h)∂h±i (mh ∓ hei )∓ = αi (mh)∓ and (α)

(α)

(mi h)∂h±i ∂h∓i (mh ∓ hei )∓ = (αi − 1)(mi h)∂h±i (mh ∓ hei )∓

we obtain by direct calculation the following formulae for homogeneous polynomials of degree k, Pk± ∈ Π± k, Bh± Pk±

= ±khPk± ,

± A± h Pk

=

± Rh,r Pk± ± Vh,r Pk±

(29)

2

kh P ±, 1±h k

kh2 = r+k− Pk± , 1±h



h kh2 = r+ 1± k− Pk± . 2 1±h

(30) (31) (32)

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Now, using the difference rules (2) and (3), we get Bh± ((mh)fh )

= (mh)Bh± fh + h1± fh (mh) + h2 Dh± fh ,

(33)

Ch± ((mh)fh )

= (mh)Ch± fh − Eh± fh ,

(34)

A± h ((mh)fh )

± = (mh)A± h fh ∓ hCh fh ,

(35)

Eh± ((mh)fh )

= (mh)Eh± fh + Ch± fh ,

(36)

± Rh,r ((mh)fh )

± = (mh)Rh,r fh + (1 ± h)Ch± fh ,

(37)

± Vh,r ((mh)fh )

± = (mh)Vh,r fh + (1 ± h)Ch± fh + 1 (h1± fh (mh) + h2 Dh± fh ), + 2

(38)

 where 1± := ± ni=1 ei . As a direct consequence of formulae (32) and (36), we obtain for the discrete homogeneous monogenic polynomials of degree k, Mk± ∈ M± k, ± Γ± h ((mh)Mk )

= −Eh± ((mh)Mk± ) − (mh)Dh± ((mh)Mk± )

2kh2 ± hk (mh)Mk± − Ch± Mk± = n+k− 1±h

(39)

by applying Theorem 3.2 and relation (16). Moreover, using identities (21), (29),(31),(32),(16) and Proposition 3.2, we obtain the relation  ±  ± ± Dh (mh) ((mh)Mk± ) = (−2Rh,n/2 + Eh± + Γ± h )((mh)Mk ) =

±hk(mh)Mk± − (2 ± 2h)Ch± Mk± .

(40)

Applying relations (36) and (34) we have Eh± ((mh)2 Mk± ) = k(mh)2 Mk± + 2(mh)Ch± Mk± − kMk± .

(41)

From Theorem 3.2 and formulae (33) and (40), we get Dh± ((mh)2 Mk± ) = −(2 ± 2h)Ch± Mk± + h1± Mk± .

(42)

Using (16) and formulae (41) and (42), we obtain ± ± ± ± ± ± 2 2 ± Γ± h ((mh) Mk ) = −k(mh) Mk ± 2(mh)Ch Mk + kMk − h1 (mh)Mk .

(43)

Applying recursively our formulae (33)-(38), it is possible to obtain explicit ± s formulae for Dh± ((mh)s Mk± ), Eh± ((mh)s Mk± ) and Γ± h ((mh) Mk ) by induction.

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3.5. Homogeneous powers Contrary to the continuous case, the classical product between the variable mh and the homogeneous polynomial Pk± is not homogeneous. However, applying formulae (6) and (7) proved in Theorem 3.2, we can say that the product (mh)Pk± can be expressed as a linear combination of homogeneous polynomials up to degree k + 1. On the other hand, the powers xs = (mh)s are not homogeneous. For this purpose, we will introduce the discrete analogues of xs in the following way: (α) Starting from the multi-index factorial powers of degree |α|, (mh)∓ , we introduce the polynomials Hs± , s ∈ N by   (2α) (−1)k k!  (mh)∓ if s = 2k, k ∈ N0  |α|=k α! Hs± (mh) =  n (−1)k k!   (2α+ei ) (mh)∓ ei if s = 2k + 1, k ∈ N0 . |α|=k i=1 α! (44) As a direct consequence of the identity  (−1)k |x|2k if s = 2k, k ∈ N0 s x = xx2k if s = 2k + 1, k ∈ N0 we can conclude by Lemma 3.3 and by the multinomial theorem, that the polynomials Hs± give rise to homogeneous polynomials of degree s which approximate the powers xs = (mh)s for small mesh width h > 0. As a direct consequence, the operator formulae proved in Subsection 3.4 are fulfilled for the powers Hs± and, moreover, by direct calculation, we obtain the additional properties ± , Ch± Hs± = Hs+1 and ± Dh± Hs± = −sHs−1 .

Let us remark that the term Ch± Hs± is the discrete version of the multiplication xxs in the continuous case.

4. A Discrete Harmonic Fischer Decomposition According to the classical theory of the finite differences, the usual approximation of the Laplacian is given by (∆h u)(mh) = =

n  u(mh + hei ) + u(mh − hei ) − 2u(mh) i=1 n  i=1

h2 (∂h∓i ∂h±i u)(mh).

(45)

The first problem that arises now is that not all of our partial difference operators do commute in the certain sense (c.f. [5, 6]) and, moreover, we have

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no factorization of the discrete Laplacian ∆h by means of our difference Dirac operators considered above, that is Dh∓ Dh± = −e0 ∆h . Let us restrict ourselves in this section to the case of quaternion-valued functions defined on lattices in R3 . Let us remark that the quaternionic variable mh is identified with the 4 × 4 matrix   0 −m1 h −m2 h −m3 h  m1 h 0 −m3 h m2 h  . mh =   m2 h m3 h 0 m1 h  0 m3 h −m2 h m1 h In [7] for a lattice function fh : R3h → H given by fh =

3 

fhi ei = fh0 e0 + Vec fh

i=0

a finite difference approximation of our Dirac operator was defined in the form   0  0 −∂h−1 −∂h−2 −∂h−3 fh −1 3 2 1     ∂ 0 −∂ ∂ f h h h   h2  Dh−+ fh =  −2  ∂ ∂h3 0 −∂h1   fh  h −3 2 1 fh3 ∂h −∂h ∂h 0

−div− h Vec fh = (46) + 0 grad− h fh + curlh Vec fh  Dh+− fh

=

= with div± h Vec fh =

3

0  ∂1  h2  ∂ h ∂h3

−∂h1 0 ∂h−3 −∂h−2

−∂h2 −∂h−3 0 ∂h−1

−div+ h Vec fh + curl− h Vec fh 3

±i 0 i=1 (∂h fh )ei

e2 ∂h±2 fh2

   

(47)

0 grad+ h fh

0 ∂h±i fhi , grad± h fh =   e1  ±1  ∂ curl± Vec f = h h  h1  f h

i=1

 0 −∂h3 fh  f1 ∂h−2   h −∂h−1   fh2 fh3 0

e3 ∂h±3 fh3

and

   .  

In the latter form one can easily see the similarity with the usual Dirac operator

−divVec f Df = . gradSc f + curlVec f

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Using the discrete identities ± div± h curlh Vec fh = 0 ± 0 curl± h gradh fh = 0

∓ ∓ ± curl± h curlh Vec fh = −∆h Vec fh + gradh div h Vec fh

we obtain the following factorization of the discrete Laplacian

−∆h fh0 Dh+− Dh−+ fh = = Dh−+ Dh+− fh . −∆h Vec fh

(48)

Now, we are able to obtain a Fischer decomposition for the discrete Dirac operators Dh−+ and Dh+− . Using the fact that

0 (α) Dh−+ (mh)+ = (α) grad− h (mh)+   0  α (mh)(α−e1 )   1  + =  (α−e )   α2 (mh)+ 2  (α−e ) α3 (mh)+ 3 as well as (α) Dh+− (mh)−

=  =

   

0 (α) + gradh (mh)− 0 (α−e ) α1 (mh)− 1 (α−e ) α2 (mh)− 2 (α−e ) α3 (mh)− 3

    

+ −+ − we can prove as in Lemma 3.4, the inclusion properties Dh+− Π+ k ⊂ Πk−1 , Dh Πk ⊂ − +− + −+ − Πk−1 and, moreover, replacing Dh by Dh and Dh by Dh in the inner product (10), we obtain the Fischer decompositions:

Theorem 4.1. Fischer decomposition for Dh−+ and Dh+− + + Let Pk− ∈ Π− k (respectively, Pk ∈ Πk ) then Pk−

=

Pk+

=

k−1  s=0 k−1  s=0

−+ (mh)s Mk−s ,

(49)

+− (mh)s Mk−s .

(50)

where each Mj−+ ( respectively, Mj+− ) denotes a homogeneous discrete monogenic −+ +− polynomial of degree j, that is, Mj−+ ∈ Π− ∈ Π+ j ∩ ker Dh (respectively, Mj j ∩ ker Dh+− ).

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From the factorization property (48), we have ± ± [(mh)2 Pk± , Q± k ]h = −[Pk , ∆h Qk ]h ,

which allows us to obtain the Fischer decomposition for the discrete Laplacian: Theorem 4.2. Fischer decomposition for ∆h Let Pk± ∈ Π± k then  ± |mh|2s Hk−2s , Pk± = 2s≤k

Hj±

denotes a homogeneous discrete harmonic polynomial of degree j, where each that is, Hj± ∈ Π± j ∩ ker ∆h . As a consequence of Theorem 4.1, we obtain Fischer decompositions which relate the discrete harmonic and the discrete monogenic polynomials. Corollary 4.1. Fischer decomposition Let Hk± ∈ Π± k ∩ ker ∆h then Hk− Hk+

=

−+ Mk−+ + (mh)Mk−1 ,

(51)

=

Mk+−

(52)

+

+− (mh)Mk−1 .

where each Mj−+ ( respectively, Mj+− ) denotes a homogeneous discrete monogenic −+ +− polynomial of degree j, that is, Mj−+ ∈ Π− ∈ Π+ j ∩ ker Dh (respectively, Mj j ∩ +− ker Dh ). To define the Euler and Gamma operator Eh+− , Γ+− and Eh−+ , Γ−+ for the h h +− −+ modified Dirac operators Dh and Dh , respectively, we start to calculate the products (mh)Dh+− fh and (mh)Dh+− fh . By straightforward calculations we obtain  −1 0    ∂h fh ∂h−2 fh0 ∂h−3 fh0 m1 h −1 1 2 1 3 1   ∂h fh ∂h fh ∂h fh   m2 h  (mh)Dh−+ fh = −   ∂ 1 f 2 ∂ −2 f 2 ∂ 3 f 2  h h h h h h m3 h ∂h1 fh3 ∂h2 fh3 ∂h−3 fh3      e1  e0 e2 e3  e2 e3    +  m1 h m2 h m3 h  fh0 +  m1 h m3 h m2 h  fh1  ∂ −1 ∂ −2 ∂ −3   ∂ −1 ∂h3 ∂h2  h h h  h   e0 e1 e3    +  −m2 h −m3 h m1 h  fh2  −∂ −2 −∂h3 ∂h1  h    e0 e1 e3   +  m3 h m2 h m1 h  fh3 . (53) 2 1   ∂ −3 ∂ ∂ h h h

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and 

(mh)Dh+− fh

   ∂h1 fh0 ∂h2 fh0 ∂h3 fh0 m1 h  ∂h1 fh1 ∂ −2 fh1 ∂ −3 fh1  h h   m2 h  = −  ∂ −1 f 2 ∂ 2 f 2 ∂ −3 f 2  h h h h h h m3 h ∂h−1 fh3 ∂h−2 fh3 ∂h3 fh3     e1  e0 e2 e3 e2 e3    0   +  m1 h m2 h m3 h  fh +  m1 h m3 h m2 h  ∂1  ∂1 ∂h2 ∂h3  ∂h−3 ∂h−2 h h    e0 e1 e3    +  −m2 h −m3 h m1 h  fh2  −∂ 2 −∂h−3 ∂h−1  h    e0 e1 e2   +  m3 h m2 h m1 h  fh3 .  ∂3 ∂h−2 ∂h−1  h

   1  fh  

(54)

Hence, we can define the difference Euler operators Eh−+ and Eh+− as (Eh−+ fh )(mh) = 

(∂h−1 fh0 )(mh + he1 ) (∂h−2 fh0 )(mh + he2 ) (∂h−3 fh0 )(mh + he3 )

  (∂ −1 f 1 )(mh + he ) (∂ 2 f 1 )(mh − he ) 1 2  h h h h    (∂h1 fh2 )(mh − he1 ) (∂h−2 fh2 )(mh + he2 )  (∂h1 fh3 )(mh − he1 )



  m1 h (∂h3 fh1 )(mh − he3 )     m2 h   (∂h3 fh2 )(mh − he3 )  m3 h 

(∂h2 fh3 )(mh − he2 )

(∂h−3 fh3 )(mh + he3 )

(∂h2 fh0 )(mh − he2 )

(∂h3 fh0 )(mh − he3 )

and (Eh+− fh )(mh) = 

(∂h1 fh0 )(mh − he1 )

  (∂ 1 f 1 )(mh − he ) (∂ −2 f 1 )(mh + he ) (∂ −3 f 1 )(mh + he ) 1 2 3  h h h h h h   −1 2  (∂h fh )(mh + he1 ) (∂h2 fh2 )(mh − he2 ) (∂h−3 fh2 )(mh + he3 )  (∂h−1 fh3 )(mh + he1 ) (∂h−2 fh3 )(mh + he2 )

(∂h3 fh3 )(mh − he3 )

and Γ+− are defined by The difference Gamma operators Γ−+ h h

    m1 h    m2 h    m3 h 

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(Γ−+ h fh )(mh) = 

(∂h−1 ∂h1 fh0 )(mh)

  (∂ −1 ∂ 1 f 1 )(mh)  h h h = h   −(∂h−1 ∂h1 fh2 )(mh)  −(∂h−1 ∂h1 fh3 )(mh)   e1  −  m1 h  ∂ −1 h

e2 m2 h ∂h−2

  e0  −  −m2 h  −∂ −2 h   e0  −  m3 h  ∂ −3 h

e3 m3 h ∂h−3

e1 −m3 h −∂h3 e1 m2 h ∂h2

−(∂h−2 ∂h2 fh1 )(mh) (∂h−2 ∂h2 fh2 )(mh)

(∂h−3 ∂h3 fh0 )(mh)



  m1 h −(∂h−3 ∂h3 fh1 )(mh)     m2 h   −3 3 2 −(∂h ∂h fh )(mh)  m3 h 

−(∂h−2 ∂h2 fh3 )(mh)

(∂h−3 ∂h3 fh3 )(mh)

   e0  e2 e3   0  f (mh) −  m1 h m3 h m2 h  −1  h  ∂  ∂h3 ∂h2 h

e3 m1 h ∂h1

e2 m1 h ∂h1

(∂h−2 ∂h2 fh0 )(mh)

   1  f (mh)  h 

   2  f (mh)  h 

   3  f (mh)  h 

and (Γ+− h fh )(mh) = 

(∂h−1 ∂h1 fh0 )(mh)

  (∂ −1 ∂ 1 f 1 )(mh)  h h h = −h    −(∂h−1 ∂h1 fh2 )(mh)  −(∂h−1 ∂h1 fh3 )(mh)   e1  −  m1 h  ∂1 h   e0  −  −m2 h  −∂ 2 h   e0  −  m3 h  ∂3 h

e2 m2 h ∂h2

e3 m3 h ∂h3

e1 −m3 h −∂h−3 e1 m2 h ∂h−2

−(∂h−2 ∂h2 fh1 )(mh) (∂h−2 ∂h2 fh2 )(mh) −(∂h−2 ∂h2 fh3 )(mh)

    e0  0   f (mh) −  m1 h  h    ∂1 h

e3 m1 h ∂h−1

e2 m1 h ∂h−1

(∂h−2 ∂h2 fh0 )(mh)

   2  f (mh)  h 

   3  f (mh).  h 

e2 m3 h ∂h−3

(∂h−3 ∂h3 fh0 )(mh)



  m1 h −(∂h−3 ∂h3 fh1 )(mh)     m2 h   −3 3 2 −(∂h ∂h fh )(mh)  m3 h  (∂h−3 ∂h3 fh3 )(mh) e3 m2 h ∂h−2

   1  f (mh)  h 

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Fischer Decomposition for Difference Dirac Operators

57

As in Subsection 3.3, we have (mh)Dh−+ = −Eh−+ − Γ−+ (respectively, h +− +− +− ± ± (mh)Dh = −Eh − Γh ) and the polynomials Pk ∈ Πk satisfy Eh−+ Pk− = kPk− , (respectively, Eh+− Pk+ = kPk+ ). Moreover, if Pk− ∈ ker Dh−+ (respectively, − − +− + + Pk+ ∈ ker Dh+− ) then we have Γ−+ h Pk = −kPk , (respectively, Γh Pk = −kPk ). −+ −+ Like in Theorem 3.2 we can prove the operator property Dh Eh = I + Eh−+ Dh−+ (respectively, Dh+− Eh+− = I + Eh+− Dh+− ). In the same way we get analogous relations to the ones presented in Subsection 3.3 and in Subsection 3.4. In addition it is also possible to define the discrete versions of the quaternionic powers (mh)s with respect to our difference Dirac operators Dh−+ and Dh+− , using a similar construction as in Subsection 3.5. At this point it would be interesting to know if the operator setting we discussed here for the quaternionic case has an equivalent operator setting in the general case of Clifford algebras. Up to know it is not known, but we will discuss it in a forthcoming paper [4].

References [1] F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Research Notes in Mathematics No. 76, Pitman, London, 1982. [2] R. Delanghe, F. Sommen, and V. Sou˘cek, Clifford algebras and spinor-valued functions, Kluwer Academic Publishers, 1992. [3] N. Faustino, K. G¨ urlebeck, A. Hommel, and U. K¨ ahler, Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl. 12, no. 6 (2006), 577–596. [4] N. Faustino, U. K¨ ahler, and F. Sommen, Difference Dirac operators in Clifford Analysis, in preparation. [5] K. G¨ urlebeck, and W. Spr¨ oßig, Quaternionic Analysis and Elliptic Boundary Value Problems, International Series of Numerical Mathematics, Volume 89, Birk¨ auser Verlag Basel, 1990. [6] K. G¨ urlebeck, and W. Spr¨ oßig, Quaternionic and Clifford calculus for Engineers and Physicists, John Wiley &. Sons, Cinchester, 1997. [7] K. G¨ urlebeck, and A. Hommel, On finite difference potentials and their applications in a discrete function theory, Math. Meth. Appl. Sci. 25 (2002), 1563–1576. [8] K. G¨ urlebeck, and A. Hommel, On finite difference Dirac operators and their fundamental solutions, Adv. Appl. Clifford Algebras, 11 (S2) (2001), 89–106. [9] G. Ren, and H. R. Malonek, Almansi-type theorems in Clifford Analysis, Math. Meth. Appl. Sci. 25 (2005), 1541–1552. [10] V. S. Ryabenkij, The Method of difference potentials for some problems of continuum mechanics (Russian), Nauka, Moscow, 1987. [11] V. Lakshmikantham, and D. Trigiante, Theory of difference equations: Numerical methods and applications, second edition, Marcel Dekker, 2002.

58 Departamento de Matem´ atica Universidade de Aveiro P-3810-193 Aveiro Portugal e-mail: [email protected] [email protected] Received: April 20, 2006 Accepted: July 3, 2006

N. Faustino and U. K¨ ahler

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