Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 251–268

ANALYTIC CLIFFORDIAN FUNCTIONS

Guy Laville and Eric Lehman

Université de Caen, CNRS UMR 6139, Laboratoire Nicolas Oresme F-14032 Caen, France; guy.laville@math.unicaen.fr, lehman@math.unicaen.fr

Abstract. In classical function theory, a function is holomorphic if and only if it is complex analytic. For higher dimensional spaces it is natural to work in the context of Clifford algebras. The structures of these algebras depend on the parity of the dimension n of the underlying vector space. The theory of holomorphic Cliffordian functions reflects this dependence. In the case of odd n the space of functions is defined by an operator (the Cauchy–Riemann equation) but not in the case of even n. For all dimensions the powers of identity (z^{n}, x^{n}) are the foundation of function theory.

I. Introduction

A complex analytic function f(z) may be defined as being locally the sum of a convergent power series f(z) = ∑_{N=1}^{∞} a_{N}z^{N−1} or as being holomorphic, that is such that

(\frac{∂}{∂x} + i\frac{∂}{∂y} )f(x + iy) = 0.

A real analytic function u(x) may be defined as being locally the sum of a convergent power series or by u(x) = f(x + i0) where f is a complex holomorphic function such that f(z̄) = \overline{f(z)}. The main difference between the complex and the real case is the existence or non-existence of a differential relation characterizing holomorphy.

We extend the definitions of analyticity and holomorphy to functions defined on Clifford algebras R_{0,n} distinguishing between the case of odd n, n = 2m + 1, and the case of even n, n = 2m. We show that the equivalence between analyticity and holomorphy still holds. The cases of odd n and even n interrelate in a way that reflects the difference between the structures of the algebras R_{0,2m} and R_{0,2m+1} . In particular the center of R_{0,2m} is R although the center of R_{0,2m+1} is R ⊕ Re_{12···2m+1} , where e_{12···2m+1} is a pseudoscalar.

2000 Mathematics Subject Classification: Primary 30G35, 15A66.

II. Notation

Let V_{n} be an anti-Euclidean vector space of dimension n. For any orthonormal basis e_{1}, …, e_{n} of V_{n} we have for all distinct i and j in {1, …, n}

e_{i}^{2} = −1  and  e_{i}e_{j} = −e_{j}e_{i}.

If I ⊂ {1, …, n} and I = {i_{1}, …, i_{k}} with i_{1} < ⋯ < i_{k} we set e_{I} = e_{i_{1}}e_{i_{2}} ⋯ e_{i_{k}}. For I = ∅, we set e_{∅} = e_{0} = 1. Then (e_{I})_{I⊂{1,...,n}} is a basis of the Clifford algebra R_{0,n} seen as a real vector space. If A = ∑_{I⊂{1,...,n}} A_{I}e_{I}, with A_{I} ∈ R, is an element of R_{0,n} we call A_{0} = A_{∅} the scalar part of A and denote it by A_{0} = S(A). Following the R − C case and also Leutwiler and Eriksson-Bique [EL2], we introduce the decomposition

R_{0,n} = R_{0,n−1} ⊕ e_{n}R_{0,n−1}.

(For convenience in our computations we have chosen e_{n}R_{0,n−1} instead of R_{0,n−1}e_{n}.) This decomposition means that given a vector e_{n} there are two maps from R_{0,n} to R_{0,n−1} , denoted ℛ and 𝒥 , such that for any A in R_{0,n} we have

A = ℛA + e_{n}𝒥 A.

We have chosen the notation ℛ and 𝒥 and the following notation for conjugation to stress the fact that for n = 1, we have R_{0,1} = C, R_{0,0} = R which yield the usual relations between C and R. If A ∈ R_{0,n}, we call the conjugate of A and denote by Ā the element of R_{0,n} defined by

Ā = ℛA − e_{n}𝒥 A.

If z is a paravector, that is an element of R ⊕ V_{n}, we have

z = z_{0} + z_{1}e_{1} + ⋯ + z_{n}e_{n} with z_{0} ∈ R, z_{1} ∈ R, …, z_{n} ∈ R.

We denote by |z| the positive real number such that |z|^{2} = z_{0}^{2} + z_{1}^{2} + ⋯ + z_{n}^{2} and by x = ℛz = z_{0} + z_{1}e_{1} + ⋯ + z_{n−1}e_{n−1} the paravector in R ⊕ V_{n−1} such that

z = x + z_{n}e_{n}  and  z̄ = x − z_{n}e_{n}.

We define z_{∗} by z_{∗} = z_{0} − z_{1}e_{1} − ⋯ − z_{n}e_{n}. Then |z|^{2} = zz_{∗} = z_{∗}z.

We introduce the differential operators

\begin{aligned}D & = \frac{∂}{∂z_{0}} + e_{1} \frac{∂}{∂z_{1}} + ⋯ + e_{n} \frac{∂}{∂z_{n}} , \\ D_{∗} & = \frac{∂}{∂z_{0}} − e_{1} \frac{∂}{∂z_{1}} − ⋯ − e_{n} \frac{∂}{∂z_{n}} and \\ Δ & = DD_{∗} = D_{∗}D. \end{aligned}

Note that Δ is the usual Laplacian.

If α = (α_{0}, α_{1}, …, α_{n}) ∈ (N ∪ {0})^{n+1} is a multi-index we denote its length by |α| = α_{0} + α_{1} + ⋯ + α_{n}. The elementary multi-index ε_{k} is defined by ε_{k} = (δ_{k0}, δ_{k1}, …, δ_{kn}) where δ_{ij} is the Kronecker symbol equal to 1 if i = j and to 0 if i ≠ j.

The order on the set of multi-indexes is the lexicographical order.

III. Analytic Cliffordian polynomials

Laville and Ramadanoff [LR1] have defined holomorphic Cliffordian polynomials for odd n. The same definitions can also be used for even n. We will see that these homogeneous polynomials are the building blocks of analytic Cliffordian functions in both cases. Therefore we call them analytic Cliffordian polynomials. Polynomials with the same structure were introduced by Heinz Leutwiler in [Le1] and also in [EL1]. Monogenic polynomials are particular cases of analytic Cliffordian polynomials [BDS] and [DSS].

III.1. Three classes of analytic Cliffordian polynomials.

Definition 1.a. Let a be a paravector and N ∈ N. We call elementary analytic monomial and denote by M_{N}^{a}(z) the homogeneous monomial function of degree N − 1 defined by

M_{N}^{a}(z) = (az)^{N−1}a = a(za)^{N−1}.

Remark 1. We may also define M_{N}^{a}(z) for all integers by M_{0}^{a}(z) = z^{−1} and for N < 0, M_{N}^{a}(z) = M_{1−N}^{z^{−1}}(a^{−1}). It is often convenient to write a(za)^{N−1} instead of M_{N}^{a}(z). We have then for N ∈ Z and \sqrt{a} a paravector such that (\sqrt{a} )^{2} = a

a(za)^{N} = \sqrt{a}(\sqrt{a} z\sqrt{a} )^{N} \sqrt{a} .

These polynomials are close to similar ones in [Le2].

Remark 2. The monomial M_{N}^{a}(z) = |a|^{N}M_{N}^{a/|a|}(z) = |z|^{N−1}M_{N}^{a}(z/|z|).

Proposition 1. We have M_{N}^{a}(z) ∈ R ⊕ V_{n} and |M_{N}^{a}(z)| = |a|^{N}|z|^{N−1} .

Proof. Computing aza explicitly, we get aza ∈ R ⊕ V_{n} and |aza| = |a|^{2}|z|. Note that

(∗)  M_{N}^{a}(z) = aM_{N−1}^{z}(a)a.

Proposition 2. We have

M_{N}^{a}(z) = D_{∗} \frac{1}{N} S((az)^{N}).

Proof. Let us define θ by |a| |z| cos θ = S(az), 0 ≤ θ ≤ π. Then 2S(az) = az + z_{∗}a_{∗} and aza = (az + z_{∗}a_{∗})a − z_{∗}a_{∗}a = 2S(az)a − |a|^{2}z_{∗} = 2|a| |z| cos θa − |a|^{2}|z|^{2}z^{−1} . A simple recursion using (∗) yields

M_{N}^{a}(z) = |a|^{N−1}|z|^{N−1} \frac{sin Nθ}{sin θ} a − |a|^{N}|z|^{N} \frac{sin(N − 1)θ}{sin θ} z^{−1}.

Since 2S((az)^{N}) = M_{N}^{a}(z)z + z_{∗}M_{N}^{a}(z)_{∗}, we get

S((az)^{N}) = |a|^{N}|z|^{N} cos(Nθ).

From the definitions of |z| and θ, we get D_{∗}|z| = |z|z^{−1} and

D_{∗}θ = \frac{cos θ}{sin θ} z^{−1} − |a|^{−1}|z|^{−1} \frac{1}{sin θ} a.

Finally

D_{∗}S(az)^{N} = N|a|^{N}|z|^{N}z^{−1} cos(Nθ) − N|a|^{N}|z|^{N} sin NθD_{∗}θ = NM_{N}^{a}(z).

Corollary. We have

z^{N} = D_{∗} \frac{1}{N + 1}S(z^{N+1}).

Proof. Choose a = e_{0} and replace N by N + 1 in Proposition 2.

Remark. Let T_{N}(x) and U_{N}(x), x ∈ R, be the classical Tchebycheff polynomials of the first and second kind. Recall that

T_{N}(x) = cos(N Arc cos x),
U_{N}(x) = \frac{sin((N + 1) Arc cos x)}{sin(Arc cos x)}.

Thus, when |a| = 1, |z| = 1, we get

S((az)^{N}) = T_{N}(S(az))  and  M_{N}^{a}(z) = U_{N−1}(S(az))a − U_{N−2}(S(az))z^{−1}.

Definition 1.b. Let a_{1}, …, a_{k} be paravectors and let N_{1}, …, N_{k} be integers belonging to N∪{0}. We set N = N_{1} +⋯+N_{k} and denote by 𝒫_{N_{1},...,N_{k}} the set of partitions I of {1, …, N} into a union of disjoint subsets I = (I_{1}, …, I_{k}) such that Card I_{1} = N_{1}, …, Card I_{k} = N_{k}. For I ∈ 𝒫_{N_{1},...,N_{k}} and ν ∈ {1, …, N}, we define b_{ν}^{I} by b_{ν}^{I} = a_{j} where j is the element of {1, …, k} such that ν ∈ I_{j}. We define the a_{1}, …, a_{k} symmetrical analytic homogeneous polynomial of degree N − 1 in z and N_{j} in a_{j} by

S_{N_{1},...,N_{k}}^{a_{1},...,a_{k}}(z) = ∑_{I∈𝒫_{N_{1},...,N_{k}}} ( ∏_{ν=1}^{N−1}(b_{ν}^{I}z))b_{N}^{I}.

Proposition 3. The polynomial S_{N_{1},...,N_{k}}^{a_{1},...,a_{k}} is a real linear combination of elementary analytic Cliffordian monomials of degree N_{1} + ⋯ + N_{k} − 1.

Proof. For any real λ, we have

M_{N}^{a+λb}(z) = ∑_{p+q=N} λ^{q}S_{p,q}^{a,b}(z).

Choose N + 1 different values for λ; one gets a Van der Monde matrix which is invertible. This shows the result for k = 2. We can iterate the same argument noting that for any real λ

S_{N_{1},...,N_{k−1},N_{k}}^{a_{1},...,a_{k−1},a_{k}+λa_{k+1}}(z) = ∑_{p+q=N_{k}} λ^{q}S_{N_{1},...,N_{k−1},p,q}^{a_{1},...,a_{k−1},a_{k},a_{k+1}}(z).

Definition 2. For each multi-index α ∈ (N ∪ {0})^{n+1} we define Q_{α} as the homogeneous polynomial in z_{0}, …, z_{n} of degree |α| − 1, by

Q_{α}(z) = ∂_{α}z^{2|α|−1}

where ∂_{α} is the differential operator of order |α|: ∂_{α} = ∂^{|α|}/∂z_{0}^{α_{0}}∂z_{1}^{α_{1}} … ∂z_{n}^{α_{n}}.

Definition 3. For each multi-index α ≠ (0, 0, …, 0) we define the analytic Cliffordian polynomial P_{α} by

P_{α}(z) = ∑_{σ∈S_{α}}( ∏_{ν=1}^{|α|−1}(e_{σ(ν)}z))e_{σ(|α|)}

where S_{α} is the set of maps σ from {1, …, |α|} to {0, 1, …, n} such that

Card(σ^{−1}({k})) = α_{k} for all k in {0, 1, …, n}.

III.2. Relations among the M_{N}^{a}(z), the Q_{α}(z) and the P_{α}(z). For any N in N, the real linear space generated by the elementary analytic monomials M_{N}^{a}, the real linear space generated by the Q_{α} with |α| = N and the real linear space generated by the P_{α} with |α| = N are identical.

Proposition 4. The polynomial Q_{α}(z) = ∂_{α}M_{2|α|}^{1}(z).

Corollary. For any multi-index α, Q_{α}(z) ∈ R ⊕ V_{n} and there exists a scalar polynomial q_{α}(z) homogeneous of degree |α| such that Q_{α}(z) = D_{∗}q_{α}(z).

Proof. ∂_{α} and D_{∗} commute and M_{2|α|}^{1}|z| = z^{2|α|−1} . Use Proposition 1 and corollary of Proposition 2.

Proposition 5. We have

Q_{α}(z) = k_{α}P_{α}(z) + ∑_{{α′>α \atop |α′|=|α|}} λ_{αα′}P_{α′}(z)

where k_{α} = \binom{2|α|−1}{α_{0}} α_{0}!α_{1}! ⋯ α_{n}! and λ_{αα′} ∈ Z.

Proof. Let β = (α_{1}, …, α_{n}) ∈ (N − {0})^{n} be the multi-index such that α = (α_{0}, β). From the definition of Q_{α} follows:

Q_{α}(z) = ( \begin{matrix}2|α| − 1 \\ α_{0} \end{matrix} ) α_{0}!\frac{∂^{|β|}}{∂z_{1}^{α_{1}} ⋯ ∂z_{n}^{α_{n}}} z^{2|β|−1+α_{0}}.

Consider z^{2|β|−1+α_{0}} as an explicit product of 2|β| − 1 + α_{0} factors each equal to z, that is: z^{2|β|−1+α_{0}} = z · z · z ⋯ z. To apply ∂/∂x_{k} is equivalent to replace each z once by e_{k} and to add all the products obtained. If we never derivate two successive z then we get α_{1}! ⋯ α_{n}!P_{α}(z). If we derivate two successive z, we get factors e_{k}e_{h} which anihilate if k ≠ h because e_{k}e_{h} = −e_{h}e_{k} and else −1 if k = h. Since 1 = e_{0} we get the terms of P_{α′} with α′ > α in the lexicographical order.

Corollary 1. The polynomial P_{α}(z) = k_{α}^{−1}Q_{α}(z) + ∑_{{α′>α \atop |α′|=|α|}} μ_{αα′}Q_{α′}(z)

where μ_{αα′} ∈ Q.

Corollary 2. For any multi-index α, P_{α}(z) ∈ R ⊕ V_{n} and there exists a scalar polynomial p_{α}(z) homogeneous of degree |α| such that P_{α}(z) = D_{∗}p_{α}(z).

Proposition 6. The monomial function M_{N}^{a}(z) = ∑_{|α|=N} a^{α}P_{α}(z), where a^{α} := a_{0}^{α_{0}}a_{1}^{α_{1}} ⋯ a_{n}^{α_{n}}.

Proposition 7. For α = (α_{0}, α_{1}, …, α_{n}), we have P_{α}(z) = S_{α_{0},α_{1},...,α_{n}}^{e_{0},e_{1},...,e_{n}}(z).

Proof. The definition of P_{α} is the definition of S in which the k paravectors a_{1} ⋯ a_{k} are the (n + 1) elements of a basis of S ⊕ V : e_{0}, e_{1}, …, e_{n}.

Corollary. The real linear space generated by the P_{α} with |α| = N is independent of the basis e_{1}, …, e_{n} of V_{n}.

Remarks. In the case of odd n, this is already known.

Note that generally the polynomials P_{α}(z) with |α| = N are not R-linearly independent. For example, if n = 3 and N = 4 we have

3P_{4000} +3P_{0400} +3P_{0040} +3P_{0004} +P_{2200} +P_{2020} +P_{2002} +P_{0220} +P_{0202} +P_{0022} = 0.

III.3. Some properties of the polynomials P_{α}.

Proposition 8. For α = (α_{0}, …, α_{n}), the polynomial p_{α}(z) is even in z_{k} if α_{k} is even and odd in z_{k} if α_{k} is odd.

Proof. We know that P_{α}(z) ∈ R ⊕ V_{n}. Then we can write

P_{α}(z) = A_{0}(z) + A_{1}(z)e_{1} + ⋯ + A_{n}(z)e_{n}.

Suppose α_{k} even. P_{α}(z) is a sum of terms like

e_{i_{1}}ze_{i_{2}}z ⋯ ze_{i_{|α|}}

in which e_{k} occurs α_{k} times, that is an even number of times. Write z = z_{0} + z_{1}e_{1}+⋯+z_{n}e_{n} and develop all the terms. The terms which contribute to A_{k}(z)e_{k} must contain an odd number of times the vector e_{k} and then z_{k}e_{k} has to appear an odd number of times. A_{k}(z) is then a sum of terms which are all odd in z_{k} and A_{k}(z) is odd in z_{k}. Since P_{α}(z) = D_{∗}p_{α}(z), we have A_{k}(z) = −(∂/∂z_{k})p_{α}(z). Since p_{α}(z) is homogeneous, we can conclude that p_{α}(z) is even in z_{k}.

If α_{k} is odd the proof is the same.

Examples. p_{(2,0,1,0)}(z) = (3z_{0}^{2} − z_{1}^{2} − z_{2}^{2} − z_{3}^{2})z_{2} is even in z_{0}, z_{1} and z_{3} and odd in z_{2} . p_{(1,1,1,0)}(z) = 8z_{0}z_{1}z_{2} is odd in z_{0}, z_{1} and z_{2} and even in z_{3} .

Corollary. If α_{n} is even, then P_{α}(z̄) = \overline{P_{α}(z)}; if α_{n} is odd then P_{α}(z̄) = −\overline{P_{α}(z)}.

Proof. Let us write

P_{α}(z) = A_{0}(z) + A_{1}(z)e_{1} + ⋯ + A_{n−1}(z)e_{n−1} + A_{n}(z)e_{n}.

If α_{n} is even, then p_{α}(z) is even in z_{n} and the polynomials

A_{k} = ±\frac{∂}{∂z_{k}} p_{α}(z)

for k ≠ n, are all even in z_{n}, but A_{n} is odd in z_{n}. Thus

P_{α}(z̄) = A_{0}(z) + A_{1}(z)e_{1} + ⋯ + A_{n−1}(z)e_{n−1} − A_{n}(z)e_{n} = \overline{P_{α}(z)}.

If α_{n} is odd, A_{0}, A_{1}, …, A_{n−1} are odd in z_{n} and A_{n} is even in z_{n}, so that

P_{α}(z̄) = −A_{0}(z) − A_{1}(z)e_{1} − ⋯ − A_{n−1}(z)e_{n−1} + A_{n}(z)e_{n} = −\overline{P_{α}(z)}.

Remark. We have

\frac{∂}{∂z_{i}} P_{α}(z) = e_{i}^{2}\{2|α|P_{α−ε_{i}}(z) − (α_{i} + 1) ∑_{k=0}^{n} P_{α+ε_{i}−2ε_{k}}(z)\}.

IV. Analytic Cliffordian functions and holomorphic Cliffordian functions

Definition 1. Let Ω be a domain of R ⊕ V_{n} and f: Ω → R_{0,n}. We say that f is a left analytic Cliffordian function if any ω in R ⊕ V_{n} has a neighbourhood Ω_{ω} in Ω such that for any z in Ω_{ω}, f(z) is the sum of a convergent series

f(z) = ∑_{N=1}^{∞} ∑_{a∈A_{N}} M_{N}^{a}(z − ω)C_{a}

where for each N in N, A_{N} is a finite subset of R ⊕ V_{n}, for each a in A_{N}, C_{a} ∈ R_{0,n} and ∑_{N=1}^{∞} ∑_{a∈A_{N}} |a|^{N}|z − ω|^{N−1}|C_{a}| is convergent in Ω_{ω}.

Remark 1. The relation M_{N}^{a}(z − ω) = ∑_{p+q=N}(−1)^{q}S_{p,q}^{a,aωa}(z) and Proposition 3 prove the consistency of the definition with respect to translations. Consequentely we will restrict ourselves to the case ω = 0.

Remark 2. The above definition is obviously intrinsic, but we get an equivalent definition if we replace the monomials M_{N}^{a} by the polynomials P_{α}; the function f: R ⊕ V_{n} → R_{0,n} is left analytic Cliffordian in a neighbourhood Ω of 0 if for every z in Ω, f(z) is the sum of a convergent series

f(z) = ∑_{N=1}^{∞} ∑_{|α|=N} P_{α}(z)c_{α}

where α are multi-indexes belonging to ({0} ∪ N)^{1+n}, and for each α we have c_{α} ∈ R_{0,n} and ∑_{N=1}^{∞} ∑_{|α|=N} |P_{α}(z)| |c_{α}| is convergent.

Definition 2. Let Ω be a domain in R ⊕ V_{n}. A function u: R ⊕ V_{n} → R_{0,n} is called a left holomorphic Cliffordian function

(i) for odd n, if DΔ^{m}u = 0 where m = \frac{1}{2}(n − 1), (ii) for even n, if for any ω ∈ Ω a neighbourhood Λ_{ω} in R ⊕ V_{n+1} and a left

holomorphic Cliffordian function f defined on Λ_{ω} exist such that – for all z in Λ_{ω}: z̄ is in Λ_{ω} and f(z̄) = \overline{f(z)},

– for all x in Λ_{ω} ∩ (R ⊕ V_{n}), u(x) = f(x).

R ⊕ V_{n+1}

Re_{n+1}

R ⊕ V_{n}

Theorem. Let Ω be a domain of R ⊕ V_{n}. A function f: Ω → R_{0,n} is left analytic Cliffordian if and only if it is left holomorphic Cliffordian.

For odd n, the theorem has already been proven in [LR1]. Let n be even, n = 2m. Let u be a left analytic Cliffordian function on a neighbourhood Ω of 0 and let S_{n}(r) be a sphere of center 0 and radius r > 0 included in Ω. For |x| < r we have

u(x) = ∑_{N=1}^{∞} ∑_{a∈A_{N}} M_{N}^{a}(x)C_{a}

where ∑_{N=1}^{∞} ∑_{a∈A_{N}} |a|^{N}|x|^{N−1}|C_{a}| is convergent. Choose Λ_{0} = S_{n+1}(r) the interior of the sphere of center 0 and radius r in R ⊕ V_{n+1} and let f: S_{n+1}(r) → R_{0,n+1} be defined by

f(z) = ∑_{N=1}^{∞} ∑_{a∈A_{N}} M_{N}^{a}(z)C_{a}.

Then f is left holomorphic Cliffordian on S_{n+1}(r) and f(x) = u(x). Since a ∈ R ⊕ V_{n}, one gets M_{N}^{a}(z̄) = \overline{M_{N}^{a}(z)}. And since C_{a} ∈ R ⊕ V_{n}, we have f(z̄) = \overline{f(z)}.

Conversely, let f be a left holomorphic Cliffordian function defined on a neighbourhood Λ_{0} of 0 in R ⊕ V_{n+1} such that f(z̄) = \overline{f(z)}. We want to show that u: Λ_{0} ∩ R ⊕ V_{n} → R_{0,n}, x ⟼ u(x) = f(x) is left analytic Cliffordian.

Since n + 1 is odd, f is analytic Cliffordian and we can write

f(z) = ∑_{N=1}^{∞} ∑_{|β|=N} P_{β}(z)c_{β} = D_{∗} ∑_{N=1}^{∞} ∑_{|β|=N} p_{β}(z)c_{β}.

Let H_{N} be the real linear space of scalar homogeneous polynomials in z_{0}, …, z_{n}, z_{n+1} of total degree N and (m + 1)-harmonic generated by (p_{β})_{|β|=N} (the dimension of H_{N} is C_{N+n}^{n} − C_{N}^{n}). We can extract a subset B_{N} of {β ∈ ({0} ∪ N)^{n+2}/|β| = N} such that (p_{β})_{β∈B_{N}} is a basis of H_{N}. D_{∗} is a linear map from H_{N} to R ⊗ V_{n+1} and Ker D_{∗} is a subspace of H_{N}. Let (ψ_{j})_{j∈𝒥} be a basis of Ker D_{∗}; since (p_{β})_{β∈B_{N}} is a basis of H_{N} there is a subset B_{N}^{∘} of B_{N} such that ((ψ_{j})_{j∈𝒥} , (p_{β})_{β∈B_{N}^{∘}} ) is a basis of H_{N}. Then (ψ_{j} ⊗ e_{I})_{j∈𝒥 ,I⊂{1,...,n+1}} , (p_{β} ⊗ e_{I})_{β∈B_{N}^{∘} ,I⊂{1,...,n+1}} is a basis of the real linear space H_{N} ⊗R_{0,n+1} , and there are unique real numbers θ_{j,I} and d_{β,I} such that

∑_{|β|=N} p_{β}(z)c_{β} = ∑_{j∈𝒥} ∑_{I⊂{1,...,n+1}} θ_{j,I}ψ_{j}(z)e_{I} + ∑_{β∈B_{N}^{∘}} ∑_{I⊂{1,...,n+1}} d_{β,I}p_{β}(z)e_{I}.

Let us write d_{β} = ∑_{I} d_{β,I}e_{I}, using homogeneity we get for any analytic Cliffordian function f the existence and unicity of the coefficients d_{β} in R_{0,n+1} such that

f(z) = ∑_{N=1}^{∞} ∑_{β∈B_{N}^{∘}} P_{β}(z)d_{β}.

Let B_{N}^{∘+} be the set of multi-indexes β = (β_{0}, …, β_{n}, β_{n+1}) where β ∈ B_{N}^{∘} and β_{n+1} is even and B_{N}^{∘−} = B_{N}^{∘} −B_{N}^{∘+} . If β ∈ B_{N}^{∘+} , the corollary of Proposition 8 implies P_{β}(z̄) = \overline{P_{β}(z)} and if β ∈ B_{N}^{∘−} we have P_{β}(z̄) = −\overline{P_{β}(z)}. The relation f(z̄) = \overline{f(z)} becomes then

∑_{N=1}^{∞}\{ ∑_{β∈B_{N}^{∘+}} \overline{P_{β}(z)}d_{β} − ∑_{β∈B_{N}^{∘−}} \overline{P_{β}(z)}d_{β}\} = ∑_{N=1}^{∞}\{ ∑_{β∈B_{N}^{∘+}} \overline{P_{β}(z)}d̄_{β} + ∑_{β∈B_{N}^{∘−}} \overline{P_{β}(g)}d̄_{β}\}.

By homogeneity we get

∑_{β∈B_{N}^{∘+}} \overline{P_{β}(z)}(d̄_{β} − d_{β}) + ∑_{β∈B_{N}^{∘−}} \overline{P_{β}(z)}(d̄_{β} + d_{β}) = 0

and by conjugation

∑_{β∈B_{N}^{∘+}} P_{β}(z)(d_{β} − d̄_{β}) + ∑_{β∈B_{N}^{∘−}} P_{β}(z)(d_{β} + d̄_{β}) = 0.

By unicity of the coefficients of the P_{β} for β ∈ B_{N}^{∘}, we get d_{β} = d̄_{β} if β ∈ B_{N}^{∘+} and d̄_{β} = −d_{β} if β ∈ B_{N}^{∘−}. Let us write a_{β} = d_{β} = dβ̄ if β ∈ B_{N}^{∘+} and e_{n+1}b_{β} = d_{β} = −d̄_{β} if β ∈ B_{N}^{∘−}, we get

f(z) = ∑_{N=1}^{∞}{ ∑_{β∈B_{N}^{∘+}} P_{β}(z)a_{β} + ∑_{β∈B_{N}^{∘−}} P_{β}(z)e_{n+1}b_{β}},

where a_{β} ∈ R_{0,n} and b_{β} ∈ R_{0,n}. The two following lemmas will then prove the theorem.

Lemma 1. Let n = 2m and β = (β_{0}, …, β_{n}, β_{n+1}). If β_{n+1} is even then the restriction of P_{β} to R ⊕ V_{n} is analytic Cliffordian from R ⊕ V_{n} to R_{0,n}.

Proof. First we show that if x ∈ R ⊕ V_{n} then P_{β}(x) ∈ R_{0,n} or better P_{β}(x) ∈ R ⊕ V_{n}. We know that P_{β}(z) = D_{∗}p_{β}(z) in R_{0,n+1} and p_{β}(z) is even in z_{n+1} since β_{n+1} is even. So

[\frac{∂}{∂z_{n+1}} p_{β}(z)]_{z_{n+1}=0} = 0

and P_{β}(x) ∈ R ⊕ V_{n} for x = ℛz.

Secondly we show that P_{β}(x) is analytic Cliffordian. We know that

P_{β}(x) = ∑_{σ∈S_{β}}(∏_{ν=1}^{n}(e_{σ(ν)}x))e_{σ(n+1)},

which means that P_{β}(x) is the sum of all different polynomials deduced from

(∗)  \underbrace{e_{0}xe_{0} ⋯ e_{0}}_{β_{0} times e_{0}} x \underbrace{e_{1}xe_{1} ⋯ e_{1}}_{β_{1} times e_{1}} xe_{2}xe_{2} · · · · · · e_{n}x \underbrace{e_{n+1}xe_{n+1}x ⋯ x_{n}e_{n+1}}_{β_{n+1} times e_{n+1}}

by permutations of the e′_{i}s. Note that

e_{n+1} = (−1)^{m}e_{1 2···n}e_{1 2···n+1}

and that the pseudo scalar e_{1 2···2m 2m+1} belongs to the center of R_{0,2m+1} . Since β_{n+1} is even and since (e_{1 2···n+1})^{2} ∈ {1, −1}, we deduce that up to a sign we can replace (∗) by:

(∗∗)  \underbrace{e_{0}xe_{0} ⋯ e_{0}}_{β_{0} times e_{0}} x \underbrace{e_{1}xe_{1} ⋯ e_{1}}_{β_{1} times e_{1}} xe_{2}xe_{2} · · · · · · e_{n}x \underbrace{e_{1 2···n}xe_{1 2···n} ⋯ xe_{1 2···n}}_{β_{n+1} times e_{1 2···n}} .

Let us choose a basis in V_{n} such that x = x_{0}e_{0} + x_{1}e_{1} . Then e_{1} commutes with x and for i ≥ 2, e_{i}e_{i+1} commutes with e_{1} , x and of course e_{0} . Suppose β_{2} > 0, for each term of the form

Ae_{2}Be_{1 2···n}C

we have another term equal to Ae_{1 2···n}Be_{n}C. But then the commutation rules give, since n = 2m,

Ae_{2}Be_{1 2···n}C + Ae_{1 e···n}Be_{2}C = 0.

Thus β_{2} = 0. And similarly β_{3} = ⋯ = β_{n} = 0. We get:

P_{β}(x) = ∑_{σ∈S_{β}}( ∏_{ν=1}^{|β|−1} E_{σ(ν)}x)E_{σ(|β|)}

where E_{0} = e_{0} , E_{1} = e_{1} and E_{n+1} = e_{1 2···n}. Commuting systematically e_{1} and e_{3}e_{4}, e_{5}e_{6}, …, e_{n−1}e_{n} from E_{n+1} to the right

P_{β}(x) = ( ∑_{σ∈S_{α}}( ∏_{ν=1}^{|α|−1} e_{σ(ν)}x)e_{σ(|α|)})(e_{1})^{β_{n+1}}(e_{3 4···n})^{β_{n+1}}

where α = (β_{0}, β_{1}, β_{n+1}, 0, …, 0) ∈ ({0} ∪ N)^{n+1} . Thus P_{β}(x) = ±P_{α}(x) with P_{α} analytic Cliffordian.

Lemma 2. Let n = 2m and β = (β_{0}, …, β_{n}, β_{n+1}). If β_{n+1} is odd then the restriction of P_{β}e_{n+1} to R ⊕ V_{n} is analytic Cliffordian from R ⊕ V_{n} to R_{0,n}.

Proof. Now p_{β}(z) is odd in z_{n+1} so

\begin{matrix}[\frac{∂}{∂z_{i}} p_{β}(z)]_{z_{n+1}=0} = 0 for i = 0, …, n, and \\ P_{β}(x) = −[\frac{∂}{∂z_{n+1}} p_{β}(z)]_{z_{n+1}=0}e_{n+1} \end{matrix}

and P_{β}(x)e_{n+1} is a scalar. For the second part of the proof we reduce the sum

P_{β}(x)e_{n+1} = ∑_{σ∈S_{β}}( ∏_{ν=1}^{|β|−1} e_{σ(ν)}x)e_{σ(|β|)}e_{n+1}

as in Lemma 1.

Corollary 1. The space of left analytic Cliffordian functions is an R-vector space and an R_{0,n} -right module, closed relatively to scalar derivations.

To deduce Corollary 2 and 3 from the above theorem, the following lemma is convenient.

Lemma 3. If v ∈ R ⊕ V_{2m}, then ∑_{i=0}^{2m} e_{i}ve_{i} = (1 − 2m)v_{∗}.

Corollary 2. If f is left analytic Cliffordian, then Df is also left analytic Cliffordian.

Proof. If n is odd let n = 2m + 1. Since f is left analytic Cliffordian it is left holomorphic Cliffordian and DΔ^{m}f = 0. But since D commutes with D and Δ we have

DΔ^{m}(Df) = D(DΔ^{m}f) = 0.

Thus Df is left holomorphic Cliffordian or left analytic Cliffordian.

If n is even let n = 2m. Let us write: x = z_{0} + z_{1}e_{1} + ⋯ + z_{2m}e_{2m} and D′ = ∑_{i=0}^{2m} e_{i}∂/∂z_{i}. Then we have D = D′ + e_{2m+1}∂/∂z_{2m+1} .

We want to show that if u is left analytic Cliffordian then D′u is also left analytic Cliffordian. In fact, we need only to show this for u(x) = M_{N}^{a}(x) for any a ∈ R ⊕ V_{2m} and any N ∈ N. A straightforward computation gives

D′M_{N}^{a}(x) = ∑_{k=1}^{N−1} ∑_{i=0}^{2m} e_{i}M_{k}^{a}(x)e_{i}M_{N−k}^{a}(x)

and since M_{k}^{a}(x)inR ⊕ V_{2m}, Lemma 3 gives us

D′M_{N}^{a}(x) = −(2m − 1) ∑_{k=1}^{N−1} [M_{k}^{a}(x)]_{∗}M_{N−k}^{a}(x).

If N is odd, let N = 2M + 1; we have

D′M_{2M+1}^{a}(x) = −2(2m − 1) ∑_{k=1}^{M} |a|^{2k}|x|^{2k−2}S((xa)^{2M−2k+1}).

If N is even, let N = 2M; we have

D′M_{2M}^{a}(x) = −(2m − 1){|a|^{2M}|x|^{2M−2} + 2 ∑_{k=1}^{M} |a|^{2k}|x|^{2k−2}S((xa)^{2M−2k})}.

The same computations in R_{0,2m+1} give

DM_{2M+1}^{a}(z) = −2(2m) ∑_{k=1}^{M} |a|^{2k}|z|^{2k−2}S((za)^{2M−2k+1})

and

DM_{2M}^{a}(z) = −(2m){|a|^{2M}|z|^{2M−2} + 2 ∑_{k=1}^{M} |a|^{2k}|z|^{2k−2}S((za)^{2M−2k})}.

Then we have for any N

D′M_{N}^{a}(x) = \frac{2m − 1}{2m}[DM_{N}^{a}(z)]_{z=x}.

Since a ∈ R_{0,2m}, we have S((za)^{N−2k}) = S((xa)^{N−2k}), and since |z̄| = |z|, we have DM_{N}^{a}(z̄) = DM_{N}^{a}(z) = \overline{DM_{N}^{a}(z)}. The theorem then proves that D′M_{N}^{a}(x) is left analytic Cliffordian.

Corollary 3. If f is left analytic Cliffordian, then D_{∗}f_{∗} is also left analytic Cliffordian.

Proof. Let us write f as

f(z) = ∑_{N=1}^{∞} ∑_{|α|=N} P_{α}(z)c_{α} = ∑_{N=1}^{∞} ∑_{|α|=N} D_{∗}p_{α}(z)c_{α}

and define the left analytic Cliffordian function f̃ by

f̃(z) = ∑_{N=1}^{∞} ∑_{|α|=N} P_{α}(z)c_{α∗}.

Then we have

D_{∗}f_{∗}(z) = (Df(z))_{∗} = ∑_{N=1}^{∞} ∑_{|α|=N} Δp_{α}(z)c_{α∗} = Df̃(z).

By Corollary 1, Df̃ is left analytic Cliffordian, thus D_{∗}f_{∗} is also left analytic Cliffordian.

V. Cauchy’s problem and boundary data

V.1. Our aim. We intend to generalize to Clifford algebras of any dimensions the method of extending real analytic functions into complex holomorphic functions. Let Ω be a domain of R ⊕ V_{2m} and u: Ω → R_{0,2m}. The Cauchy problem

\{ \begin{matrix}DΔ^{m}f = 0, \\ f|_{Ω} = u \end{matrix}

where the unknown function f has to be defined on an open set Λ containing Ω, seems not well defined since the partial differential equations are of order 2m + 1. The Cauchy–Kowalewski theorem tells us that we need the normal derivatives of f to R ⊕ V_{2m} up to the order 2m. We will see that the algebraic and analytic properties of f and u enable us to compute these derivatives uniquely given the function u.

We will denote by 𝒜_{2m} the linear space of left analytic Cliffordian functions defined on Ω and taking their values in R_{0,2m}. Similarly, 𝒜_{2m+1} is the linear space of left analytic Cliffordian functions defined on Λ and with values in R_{0,2m+1} .

V.2. The operator (A | ▽_{n}).

Definition. Let A ∈ R_{0,n}. We define (A | ▽_{n}): 𝒜_{n} ⟶ 𝒜_{n} by:

(i) for all a ∈ R ⊕ V_{n} and for all N ∈ N we have

(A | ▽_{n})M_{N}^{a}(z) = ∑_{k=1}^{N−1} M_{k}^{a}(z)AM_{N−k}^{a}(z),

(ii) for all f ∈ 𝒜_{n} and for all K ∈ R_{0,n}, (A | ▽_{n})(f(z)K) = (A | ▽_{n})f(z))K, (iii) (A | ▽_{n}) is R-linear and continuous.

Consequence. If f(z) = ∑_{N=1}^{∞} ∑_{a∈A_{n}} M_{N}^{a}(z)C_{a}, then

(A | ▽_{n})f(z) = ∑_{N=1}^{∞} ∑_{a∈A_{N}}((A | ▽_{n})M_{N}^{a}(z))C_{a}.

Proposition 9. If f ∈ 𝒜_{n}, then (∂/∂z_{k})f(z) = (e_{k} | ▽_{n})f(z).

Proof. We need only to verify the proposition for f(z) = M_{N}^{a}(z). We have

\begin{aligned}\frac{∂}{∂z_{k}} azaz ⋯ za & = ae_{k}az ⋯ za + azae_{k} ⋯ za + ⋯ + azaz ⋯ e_{k}a \\ & = (e_{k} | ▽_{n})azaz ⋯ za. \end{aligned}

Proposition 10. If f ∈ 𝒜_{n} and if λ belongs to the center of R_{0,n} then

(λA | ▽_{n})f = λ(A | ▽_{n})f.

Lemma. If v ∈ R ⊕ V_{2m}, then

∑_{i=0}^{2m} e_{i}ve_{i} = (−1)^{m}(1 − 2m)e_{1 2···2m}ve_{1 2···2m}.

Proof. Both sides of the equality are equal to (1 − 2m)v_{∗}.

Proposition 11. If u ∈ 𝒜_{2m}, then

e_{1 2···2m}(e_{1 2···2m} | ▽_{2m})u = \frac{(−1)^{m+1}}{2m − 1} Du.

Proof. We need only to verify the relation for u(x) = M_{N}^{a}(x). Using the definition and the lemma, we get

(−1)^{m}(1 − 2m)e_{1 2···2m}(e_{1 2···2m} | ▽_{2m})u = ∑_{k=1}^{N−1} ∑_{i=0}^{2m} e_{i}M_{k}^{a}(x)e_{i}M_{N−k}^{a}(x)

= ∑_{i=0}^{2m} e_{i} ∑_{k=1}^{N−1} M_{k}^{a}(x)e_{i}M_{N−k}^{a}(x)

= ∑_{i=0}^{2m} e_{i} \frac{∂}{∂x_{i}} M_{N}^{a}(x) = DM_{N}^{a}(x).

Proposition 12. For μ ∈ {0} ∪ N, we have

\begin{aligned}(e_{1 2···2m} | ▽_{2m})^{2μ}u & = \frac{(−1)^{mμ}}{(2m − 1)^{2μ}} Δ^{μ}u, \\ (e_{1 2···2m} | ▽_{2m})^{2μ+1}u & = \frac{(−1)^{1+mμ}}{(2m − 1)^{2μ+1}} e_{1 2···2m}DΔ^{μ}u. \end{aligned}

Proof. From Proposition 3, we get

(e_{1 2···2m} | ▽_{2m})u = \frac{−1}{2m − 1}e_{1 2···2m}Du = \frac{−1}{2m − 1}D_{∗}u_{∗}e_{1 2···2m}.

Corollary 2 of Section IV shows that D_{∗}u_{∗}e_{1 2···2m} is left analytic Cliffordian. Thus we may apply the operator as many times as we want. We get

((e_{1 2···2m} | ▽_{2m})^{2}u = \frac{−1}{2m − 1}e_{1 2···2m}D\{\frac{−1}{2m − 1}D_{∗}u_{∗}e_{1 2···2m}\}

= \frac{1}{(2m − 1)^{2}} e_{1 2···2m}Δu_{∗}e_{1 2···2m}

= \frac{1}{(2m − 1)^{2}} (e_{1 2···2m})^{2}Δu = \frac{(−1)^{m}}{(2m − 1)^{2}} Δu.

A simple recursion gives then the general formulae.

V.3. Analytic extension.

Theorem. Let Ω be a domain of R ⊕ V_{2m} and let u: Ω → R_{0,2m} be left analytic Cliffordian. There exist a domain Λ in R ⊕ V_{2m+1} with Ω ⊂ Λ and a unique left holomorphic Cliffordian function f defined on Λ ⊂ R ⊕ V_{2m+1} , such that f|_{Ω} = u. That function f is such that f(z̄) = \overline{f(z)} and if we denote by ∂/∂n the normal derivative to Ω, we have for any μ in {0} ∪ N

(\frac{∂}{∂n})^{2μ}f |_{Ω}= \frac{(−1)^{μ}}{(2m − 1)^{2μ}} Δ^{μ}u, (\frac{∂}{∂n})^{2μ+1} |_{Ω}= \frac{(−1)^{μ+1}}{(2m − 1)^{2μ+1}} e_{2m+1}DΔ^{μ} u.

Proof. 1^{∘} Suppose f is a solution. We use the usual notation

z = z_{0} + z_{1}e_{1} + ⋯ + z_{2m}e_{2m} + z_{2m+1}e_{2m+1} = x + z_{2m+1}e_{2m+1}.

Thus we have

(\frac{∂}{∂n})^{j}f = (\frac{∂}{∂z_{2m+1}} )^{j}f.

Since f is left analytic Cliffordian, we have

\frac{∂}{∂z_{2m+1}} f = (e_{2m+1} | ▽_{2m+1})f

and thus

(\frac{∂}{∂n})^{j}f = (e_{2m+1} | ▽_{2m+1})^{j}f.

Note that e_{2m+1} = (−1)^{m}e_{1 2···2m2m+1}e_{1 2···2m} and that (−1)^{m}e_{1 2···2m2m+1} belongs to the center of R_{0,2m+1} . Using Proposition 10 we get

(\frac{∂}{∂n})^{j}f = ((−1)^{m}e_{1,2···2m2m+1})^{j}(e_{1 2···2m} | ▽_{2m+1})^{j}f.

Taking the restriction to z = x ∈ Ω, we get

(\frac{∂}{∂n})^{j}f|_{Ω}(x) = (−1)^{mj}(e_{1 2···2m2m+1})^{j}(e_{1 2···2m} | ▽_{2m})^{j}u(x).

Proposition 12 gives us then for j = 2μ and j = 2μ + 1

(\frac{∂}{∂n})^{2μ}f|_{Ω}(x) = \frac{(−1)^{μ}}{(2m − 1)^{2μ}} Δ^{μ}u(x)

and

(\frac{∂}{∂n})^{2μ+1}f|_{Ω}(x) = \frac{(−1)^{μ+1}}{(2m − 1)^{2μ+1}} e_{2m+1}DΔ^{μ}u(x).

2^{∘} Using the above conditions for j ≤ 2m, the theorem of Cauchy–Kowalewski proves the existence of Λ′ ⊂ R ⊕ V_{2m+1} with Λ′ ∩ (R ⊕ V_{2m}) = Ω and the existence and unicity of f in Λ′.

3^{∘} Knowing the existence of f, the usual Taylor formula gives us

\begin{aligned}f(z) & = ∑_{μ=0}^{∞} \frac{(−1)^{μ}}{(2m − 1)^{2μ}(2μ)!}(z_{2m+1})^{2μ}Δ^{μ}u(x) \\ & + e_{2m+1} ∑_{μ=0}^{∞} \frac{(−1)^{μ+1}}{(2m − 1)^{2μ+1}(2μ + 1)!}(z_{2m+1})^{2μ+1}DΔ^{μ}u(x). \end{aligned}

This formula shows, since Du(x) ∈ R_{0,2m}, that f(z̄) = \overline{f(z)} in a subdomain Λ of R ⊕ V_{2m+1} such that Ω ⊂ Λ ⊂ Λ′.

VI. Fueter’s method

Fueter’s method is well known and widely used to construct functions connected to monogenic functions [Fu], [De], [Qi]. It is known to be effective to construct holomorphic Cliffordian functions in the case of odd n. We show that it is still valid for our definition in the case of even n.

Theorem. Let φ be a complex holomorphic function defined on D_{φ} an open subset of the upper half-plane and let p and q be the real functions of two variables defined by

∀ ζ = xi + iη ∈ D_{φ}, φ(ζ) = p(ξ, η) + iq(ξ, η).

Let z⃗ = z_{1}e_{1} + ⋯ + z_{n}e_{n}, for z_{0} + i|z⃗ | ∈ D_{φ} we define u(z_{0} + z⃗) by

u(z_{0} + z⃗) = p(z_{0}, |z⃗|) + \frac{z⃗}{|z⃗ |}q(z_{0}, |z⃗ |).

Then u is a (left and right) holomorphic Cliffordian function.

Proof. For odd n, this result is already known [LR1].

If n is even, let n = 2m. Let x = z_{0} + z⃗ and z = x + z_{2m+1}e_{2m+1} . Define f by

f(z) = p(z_{0}, |z⃗ + z_{2m+1}e_{2m+1}|) + \frac{z⃗ + z_{2m+1}e_{2m+1}}{|z⃗ + z_{2m+1}e_{2m+1}|}q(z_{0}, |z⃗ + z_{2m+1}e_{2m+1}|).

From the case of odd n, we know that f is a left and right holomorphic Cliffordian function. The theorem of Section IV shows then that u is a left and right holomorphic Cliffordian function, since we have f(z̄) = \overline{f(z)} and u(x) = f(x).

268 Guy Laville and Eric Lehman

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Received 6 May 2002