Codimension-p Paley–Wiener theorems

c 2007 by Institut Mittag-Leffler. All rights reserved

Codimension-p Paley–Wiener theorems

Yan Yang, Tao Qian and Frank Sommen

Abstract. We obtain the generalized codimension-p Cauchy–Kovalevsky extension of the exponential functione^iy,tinR^m=R^p⊕R^q, wherep>1, y, t∈R^q, and prove the corresponding codimension-pPaley–Wiener theorems.

0. Introduction

The Clifford algebra formulation of Euclidean spaces will be adopted. Let e₁, ...,e_m be basic elements satisfyinge_ie_j+e_je_i=−2δ_ij, whereδ_ij=1 if i=j and δ_ij=0 otherwise,i, j=1,2, ..., m.Set

R^m={x=x₁e₁+...+x_me_m:x_j∈R, j= 1,2, ..., m}, and

R^m₁ ={x=x₀+x:x₀∈R, x∈R^m}.

R^m and R^m₁ are called the homogeneous and inhomogeneous Euclidean spaces, respectively.

A number of generalizations of the classical Paley–Wiener theorem (PW the- orem) to higher dimensional spaces were studied [1], [3], [6], [7] and [9]. Among the literature, by imbedding R^mintoC^m=R^m⊕iR^m, the corresponding PW the- orem is obtained in [6]. By making use of Clifford algebra a direct proof of the theorem is obtained in [9]. In [7], a different type of PW theorem inC^mis proved by using the heat kernel. In [3], through imbedding R^minto R^m₁ , in the complex structure induced by the generalized Cauchy–Riemann operator, an inhomogeneous codimension-1 result is proved. This latter result is viewed as a precise analogy to the classical result in whichRis imbedded into the complex planeC=R¹₁.

The first author was supported by Research Grant of University of Macau No. RG079/04- 05S/QT/FST. The third author was supported in part by “FWO-Krediet aan Navorsers:

1.5.065.04”

The standard CK extension (Cauchy–Kovalevsky extension) ([1] and [2]) as- serts that any real-analytic function in an open set Q of R^q may be extended to become a one-sided-monogenic function in an open set ofR^q₁that containsQ.This will be regarded as the inhomogeneous codimension-1 CK extension. The authors of [2] further obtain thehomogeneous codimension-pCK extension that extends any real-analytic function in an open setQof R^q into a one-sided-monogenic function in an open set of the homogeneous spaceR^p⊕R^q containing the setQ.Whenp=1 this reduces to the homogeneous codimension-1 CK extension from R^q to R^q+1. The codimension-pCK extension is made more general in [2] through incorporating ak-left-inner monogenic weight function inR^p, and calledgeneralized CK extension or homogeneous codimension-pCK extension with ak-monogenic function. In this paper, with homogeneous codimension-p and generalized CK extensions of e^iy,t from R^q intoR^p⊕R^q we prove the corresponding PW theorems inR^m=R^p⊕R^q, m=p+q. We further extend the PW theorem in terms of generalized Taylor series.

In Section 1 we recall the basic notation and terminology used in the paper.

This section also serves as a survey on the theory of CK type and “generalized- CK type” extensions of real-analytic functions, as well as generalized Taylor series.

In Section 2 we obtain the homogeneous codimension-pgeneralized CK extension of e^iy,t in R^m=R^p⊕R^q, where y, t∈R^q. In Section 3 we formulate and prove the corresponding codimension-pPW theorems. Furthermore, we obtain the PW theorems in terms of monogenic extensions given by generalized Taylor series. The proofs of the results in this work are based on the inhomogeneous codimension-1 PW theorem obtained in [3].

1. Preliminaries

The reader is supposed to know the basic material on Clifford algebras, Dirac operators, Cauchy–Riemann operator and so on. The basic knowledge and notation in relation to Clifford algebras are referred to [1], [2] and [4].

Theunit sphere {x∈R^m:|x|=1} is denoted byS^m−1. We use B(x, r) for the open ball inR^mcentered atxwith radiusr, andB(x, r) for the topological closure ofB(x, r).

Let k∈N, where N denotes the set of non-negative integers. Denote by M⁺(m, k,C^(m)) the space ofk-homogeneous monogenic polynomials in R^m, called k-monogenic of degree k.

Below by saying that f is analytic at a certain point we mean that f can be expanded into a Taylor series in a neighborhood of the point. Let f(x) be a given analytic function inU⊂R^q.When we say thatf^∗ isan inhomogeneous CK extension of f, we mean that there exist an open set U^∗ of R^q₁ containing U and

a functionf^∗, left-monogenic with respect to the inhomogeneous Dirac operator∂_x, such thatf^∗|x0=0=f in U.By ahomogeneous CK extension off toR^q+1we mean a functionf^∗, left-monogenic with respect to the homogeneous Dirac operator ∂_x in an open setU^∗ inR^q+1 containingU, such that f^∗|_x0=0=f inU. The existence of such CK extensions are referred to [1] and [2].

The Fourier transform of functions inR^mis defined by fˆ(ξ) =

R^m

e^−ix,ξf(x)dx and the inverse Fourier transform by

ˇ

g(x) = 1 (2π)^m

R^m

e^ix,ξg(ξ)dξ, whereξ=ξ₁e₁+...+ξ_me_m.

To extend the domain of the Fourier transform toR^m₁, we first need to extend the exponential functione^ix,ξ.Set, forx=x₀e₀+x,

e(x, ξ) =e^ix,ξe^−x0|ξ|χ+(ξ)+e^ix,ξe^x0|ξ|χ−(ξ), (1)

where

χ_±(ξ) =1 2

1±iξe₀

|ξ|

.

It is easy to verify that the functionsχ_± satisfy the properties of projections:

χ₋χ₊=χ₊χ₋= 0, and χ²_±=χ_±, χ₊+χ₋= 1.

For any fixed ξ, e(x, ξ) is two-sided-monogenic in x∈R^m₁. The above extension is the inhomogeneous codimension-1 CK extension ofe(x, ξ) toR^m₁. Replacinge₀by e_m+1 in (1), wheree_m+1 is a basis element added to the collectione₁, ...,e_m, with e²_m+1=−1 and anti-commutativity with the other e_j, j=1, ..., m, one obtains the homogeneous codimension-1 CK extension of e^ix,ξ in R^m+1. Generalizations of the exponential function of these types can be first found in F. Sommen’s work [5].

We will be concerned with the direct sum decomposition of the spaceR^minto R^p⊕R^q, whereR^pis the real-linear span ofe₁, ...,e_p, andR^qis the real-linear span ofe_p+1, ...,e_m.We callR^q thecodimension-pspacein R^p⊕R^q.

Codimension-pCK extensionconcerns the following question. Suppose we are given a functionf(y) that is analytic in an open subsetU⊂R^q. Then the question is: Is there a left-monogenic functionf^∗(x, y) in some domain inR^p⊕R^qcontaining U such thatf^∗|x=0=f inU?

The answer to this question is positive, but, when p>1, the solution is not unique. For instance,f(y) can first have a homogeneous codimension-1 CK exten- sion inR^q+1which is also left-monogenic in R^p⊕R^q.

Before answering the question in full, we first need to distinguish different types of monogenic extensions. In both the homogeneous and inhomogeneous codimension-1 cases there will be only one type, viz. the standard CK type, as the extension is unique. The CK extension fromU⊂R^q toU^∗⊂R^q₁corresponds to the series expansion of

f(x₀, y) =e^−x0∂yA(y) = ∞ j=0

(−1)^j

j! x^j₀∂_y^jA(y),

where ∂_y is the Dirac operator in the homogeneous Euclidean spaceR^q.

The codimension-p, p>1, CK extension of a function A(y), y∈U⊂R^q, being the case k=0 in Lemma 1 ([2]), is a modification of the series obtained from the exponential expression e^−x0∂yA(y). Lemma 1 answers the question in a more gen- eral context, called a generalized CK extension, where an extension is related to a k-monogenic function in M⁺(p, k,C^(p)). The participation of the k-monogenic function makes the extension having the role of monomial functionsz^k in one com- plex variable, which enables one to further formulate generalized Taylor series.

Lemma 1. (The Generalized CK Extension: Homogeneous Codimension-p CK extension Associated With P_k [2, p. 265])Let P_k∈M⁺(p, k,C^(p))be given and A₀(y) be a Clifford C^(q)-valued analytic function inU. Then there exists a unique sequence (A_l(y))_l>0 of analytic functions such that the series

f_Pk(x, y) = ∞

l=0

x^lP_k(x)A_l(y) (2)

is convergent in an SO(p)-invariant domainU^∗⊂R^m, which is a neighborhood ofU, and the sum f is left-monogenic in U^∗. The functions A_l(y), l>1, are uniquely determined by the formulas

P_k(x)A_2l(y) =

(−1)^lΓ(k+p/2)∂_y^2l[P_k(x)A₀(y)]

2^2ll!Γ(l+k+p/2) , and

P_k(x)A_2l+1(y) =

(−1)^lΓ(k+p/2)∂_y^2l+1[P_k(x)A₀(y)]

2^2l+1l!Γ(l+k+p/2+1) .

We callf_Pk(x, y) thegeneralized CK extension in relation to P_k of A₀(y) and A₀(y) theinitial value off_Pk(x, y). Denote byTPk(U) the space of all functions of the form (2).

From [2] we know, if f(x, y), x=ρω∈R^p, y∈R^q, is left-monogenic in R^m= R^p⊕R^q, thatT_k(f)(ω, y)=lim_ρ!0ρ^−kP(k)f(ρ, ω, y) is the generalized Taylor coef-

ficient off of order k, which can be decomposed in a unique way as T_k(f)(ω, y) =

α∈Ik

P_k,α(ω)T_k,α(f)(y),

whereT_k,α(f)(y) are real-analytic functions.

Denote the generalized CK extension, corresponding toP_k,α(ω)T_k,α(f)(y), by T_k,α(x, y), then we have

f(x, y) = ∞ k=0

α∈Ik

T_k,α(x, y), (3)

where

T_k,α(x, y) = ∞ l=0

x^lP_k,α(x)T_k,α^(l)(y).

The expansion (3) is called ageneralized Taylor series.

2. The generalized CK extension ofe^iy,t in R^m=R^p⊕R^q

In [1] and [4], the homogeneous codimension-1 CK extension ofe^iy,t is given by

e(x₁e₁, y, t) =e^iy,t

cosh(x₁|t|)+isinh(x₁|t|)e₁ t

|t|

.

It is easy to see that for ally,t∈R^q,|e(x₁e₁, y, t)|≤Ce^|x1||t|, whereC is a constant.

Now we study the extension for all casesp≥1 andk≥0, forP_k∈M⁺(p, k,C^(p)) given. Let x=rω∈R^p. In Lemma 1, settingA₀(y)=e^iy,t, we obtain the general- ized, homogeneous codimension-pCK extension of e^iy,t in TPk, denoted by ε^p_P

k, with the expression

ε^p_P

k(x, y, t) = Γ k+p

2

e^iy,tr^k r|t|

2

−k⁻p/2⁺1

(4)

×

P_k(ω)I_k+p/2−1(r|t|)+iωP_k(ω)I_k+p/2(r|t|) t

|t|

,

where

I_v(x) =i^−vJ_v(ix) = ∞ k=0

1 k!Γ(v+k+1)

x 2

_v+2k

is a kind of Bessel function ([8]). Note thate(x₁e₁, y, t)=ε¹₁(x₁e₁, y, t).

Next we will estimateε^p_P

k(x, y, t).

From [8] we know that the generating function ofI_n is

e^u(t+1/t)/2= ∞ n=⁻∞

tⁿI_n(u).

Takingt=1, we gete^u= ^∞_n=−∞I_n(u).

UsingI_n(−u)=I_n(u), we gete^u=2 ^∞_n=1I_n(u)+I₀(u). AsI_n(u)>0 whenu>0, we have

∞ n=0

I_n(x)≤e^x, whenx >0.

(5)

When|t|≤Ω, using (4) and (5) we have

|ε^p_P

k(x, y, t)| ≤C 2

Ω _k

rΩ 2

−p/2⁺1

[I_k+p/2−1(rΩ)+I_k+p/2(rΩ)]

≤C[I_k(rΩ)+I_k+1(rΩ)]≤Ce^rΩ, (6)

where Cis a constant independent ofy.

On the other hand, whenk=0 and P_k=1 we have the generalization ofe^iy,t in T₁(R^q),

ε^p₁(x, y, t) = Γp 2

e^iy,t r|t|

2

−p/2+1

I_p/2−1(r|t|)+iI_p/2(r|t|)ω t

|t|

.

Since for anyu>0, u

2 −v

I_v(u) = ∞ k=0

1 k!Γ(v+k+1)

u 2

_2k

≤C ∞ k=0

u^2k

(2k)!≤Ce^u, and

u 2

−v⁺1

I_v(u) = ∞ k=0

1 k!Γ(v+k+1)

u 2

_2k+1

≤C ∞ k=0

u^2k+1

(2k+1)!≤Ce^u. We therefore have

|ε^p₁(x, y, t)| ≤Ce^r|t| for anyy, t∈R^q, (7)

whereC is a constant independent ofy.The estimate is stronger than (6) where it is under the restriction|t|≤Ω.

3. Paley–Wiener theorems in R^m=R^p⊕R^q

We first study the codimension-1 case. The following result is obtained in [3].

Lemma 2. (Inhomogeneous codimension-1 PW theorem) Let F be analytic, defined in R^q, taking values in C^(q), the complex Clifford algebra generated by e₂, ...,e_q+1, and F∈L²(R^q). Let Ω be a positive real number. Then the following two conditions are equivalent:

(1)F can be left-monogenically extended to a function defined onR^q₁, denoted by f, and there exists a constantC such that

|f(y)| ≤Ce^Ω|y| for anyy∈R^q₁. (2) supp(F) ⊂B(0,Ω).

Moreover, if one of the above conditions holds, then we have

f(y) = 1 (2π)^q

R^q

e(y, ξ)F(ξ) dξ, y∈R^q₁,

wheree(y, ξ)is given in(1) withe₀=1.

With the differencee²₁=−1 in place ofe²₀=1 in the above result, we now prove the following result.

Theorem 1. (Homogeneous codimension-1 PW theorem)Let F be analytic, defined in R^q, taking values in C^(q), the complex Clifford algebra generated by e₂, ...,e_q+1, and F∈L²(R^q), and let Ω be a positive real number. Then the fol- lowing two conditions are equivalent:

(1)F can be left-monogenically extended to a function defined onR^q+1, denoted by f, and there exists a constantC such that

|f(x₁e₁, y)| ≤Ce^Ω|x1e1+y| for any x₁∈Randy∈R^q. (2) supp(F) ⊂B(0,Ω).

Moreover, if one of the above conditions holds, then we have

f(x₁e₁, y) = 1 (2π)^q

R^q

e(x₁e₁, y, t)F(t)dt, x₁∈R, y∈R^q.

While the proof of Lemma 2 ([3]) may be adapted step by step to give a proof of Theorem 1, we, however, prefer to show that the main part of the theorem may be concluded from Lemma 2.

Proof. (2)⇒(1) Let

G(x₁e₁, y) = 1 (2π)^q

R^q

e(x₁e₁, y, t)F(t)dt.

Because supp(F)⊂B(0,Ω), we have G(x₁e₁, y) = 1

(2π)^q

B(0,Ω)

e(x₁e₁, y, t)F(t)dt.

The estimate ofε¹₁(x₁e₁, y, t) implies

|G(x₁e₁, y)| ≤CF₂e^Ω|x1|χ_B(0,Ω)2≤Ce^Ω|x1e1+y|.

SinceF(y) andG(0, y) agree inR^q, both being left-monogenic inR^q+1, we conclude that f(x₁e₁, y)=G(x₁e₁, y).

(1)⇒(2) Since f(x₁e₁, y) is the homogeneous codimension-1 CK extension of f(0, y)=F(y), and

|f(x₁e₁, y)| ≤Ce^Ω|x1e1+y|, x₁e₁∈R¹, y∈R^q, we have

|f(x₁e₁, y)|= ^∞

l=0

Γ₁

2

|x₁|^2l∂_y^2l[A₀(y)]

(2l)! +

∞ l=0

Γ₁

2

|x₁|^2lx₁e₁∂_y^2l+1[A₀(y)]

(2l+1)!

≤Ce^Ω|x1e1+y|, where f(0, y)=A₀(y).

Since the first and the second summations are expanded, respectively, over different groups of reduced products of the basis elements of R¹⊕R^q, we have

^∞

l=0

Γ₁

2

|x₁|^2l∂_y^2l[A₀(y)]

(2l)!

≤Ce^Ω|x1e1+y|, and

^∞

l=0

Γ₁

2

|x₁|^2lx₁e₁∂_y^2l+1[A₀(y)]

(2l+1)!

≤Ce^Ω|x1e1+y|.

On the other hand,F(y) has an inhomogeneous codimension-1 CK extension f₁(x₁, y)∈R^q₁ withf₁(0, y)=F(y) and

f₁(x₁, y) = ∞

l=0

|x₁|^2l∂_y^2l[A₀(y)]

(2l)! −

∞ l=0

|x₁|^2lx₁∂_y^2l+1[A₀(y)]

(2l+1)! .

Then the last two inequalities imply that

|f₁(x₁, y)| ≤ ^∞

l=0

|x₁|^2l∂_y^2l[A₀(y)]

(2l)!

+ ^∞

l=0

|x₁|^2lx₁∂^2l_y⁺¹[A₀(y)]

(2l+1)!

≤Ce^Ω|x1e1+y|=Ce^Ω|x1+y|.

Invoking Lemma 2, we conclude that supp(F)⊂B(0,Ω).

Below we study the homogeneous codimension-pPW theorem in R^m=R^p⊕ R^q (for p>1) for CK extensions involving an initial value function. We need the following lemma.

Lemma 3. Let

∂xⁿ=∂xⁿ₁¹...∂xⁿ_l^l, |n|=n₁+n₂+...+n_l, wheren=(n₁, ..., n_l)∈N^l is an l-dimensional multi-index. Let

E(x) = x

|x|^l, x∈R^l.

Then

∂^|n|

∂xⁿE(x)

≤(l−1)l...(l+|n|−2)

|x|^l+|n|−1 .

The lemma can be easily proved by direct computation.

Lemma 4. (Technical lemma)Assume thatf(x, y)is left-monogenic inR^p⊕R^q, and

|f(x, y)| ≤Ce^Ω|x|, x∈R^p, y∈R^q,

whereΩis a positive real number. Letf₁(x₁e₁, y)be the homogeneous codimension-1 CK extension satisfyingf₁(0, y)=f(0, y).Then for anyε>0, there exists a constant C_ε>0such that

|f₁(x₁e₁, y)| ≤C_εe^(Ω+ε)|x1|, x₁e₁∈R¹, y∈R^q.

Proof. Since f(x, y) is left-monogenic inR^m=R^p⊕R^q, by Cauchy’s formula, we have

f(x, y) = 1 ω_m−1

∂B(x⁺y,ρ)

E(t−(x+y))dσ(t)f(t), whereρis any positive real number.

Then

f(0, y) = 1 ω_m−1

∂B(y,ρ)

E(t−y)dσ(t)f(t).

By Lemma 3 and the condition|f(x, y)|≤Ce^Ω|x|, we have

|∂_y^lf(0, y)| ≤C(m−1)m...(m+l−2) ρ^l e^Ωρ. (8)

Sincef₁ is the homogeneous codimension-1 CK extension andf(0, y)=f₁(0, y), the above estimate (8) implies that

|f₁(x₁e₁, y)| ≤^∞

l=0

|x₁|^l|∂_y^l[f(0, y)]|

l! ≤^∞

l=0

|x₁|^l(m−1)m...(m+l−2) l!ρ^l e^Ωρ. (9)

For anyε>0, setl₀=[2Ω(m−2)/ε]+1. Thenl>l₀will imply that (m−2)/l<ε/2Ω.

Thus, ifl>l₀, then 1+(m−2)/l<1+ε/2Ω, and from (9) we get

|f₁(x₁e₁, y)| ≤C ∞ l=0

|x₁|^l(m−1)m...(m+l−2)

l!ρ^l e^Ωρ

≤C

l0

l=0

|x₁|^l(m−1)m...(m+l−2)

l!ρ^l e^Ωρ+C_ε ∞ l=l0

|x₁|^l(1+ε/2Ω)^l−l0 ρ^m+l−1 e^Ωρ, where C_ε=(m−1)(m/2)...(m+l₀−2)/l₀.

Takingρ=|x₁|(1+ε/Ω), we have

|f₁(x₁e₁, y)| ≤C_εe^(Ω+ε)|x1|. The proof is complete.

Theorem 2. (Homogeneous codimension-pPW Theorem)Let F be analytic, defined in R^q, taking values in C^(q), the complex Clifford algebra generated by e_p+1, ...,e_p+q, and F∈L²(R^q), and let Ω be a positive real number. Then the fol- lowing two assertions are equivalent:

(1)F has a homogeneous codimension-pCK extension toR^p+q, denoted by f, and there exists a constant C such that

|f(x, y)| ≤Ce^Ω|x| for anyx∈R^p andy∈R^q. (2) supp(F)⊂B(0,Ω).

Moreover, if one of the above conditions holds, we have

f(x, y) = 1 (2π)^q

R^q

ε^p₁(x, y, t)F(t)dt for any x∈R^p andy∈R^q.

Proof. (2)⇒(1) Set

G(x, y) = 1 (2π)^q

R^q

ε^p₁(x, y, t)F(t) dt.

As supp(F) ⊂B(0,Ω), we have G(x, y) = 1

(2π)^q

B(0,Ω)

ε^p₁(x, y, t)F(t)dt.

(10)

Owing to the estimate (7) ofε^p₁(x, y, t), we have

|G(x, y)| ≤CF2e^Ω|x|χ_B(0,Ω)2≤Ce^Ω|x| for anyx∈R^pandy∈R^q. Since ε^p₁ is the codimension-p CK extension ofe^iy,t, through the integral repre- sentation (10), the functionG(x, y) is the codimension-pCK extension ofF(y).As bothG(x, y) andf(x, y) are codimension-pCK extensions from the same function F(y) onR^q, they have to be identical onR^p+q.

(1)⇒(2) SetA₀(y)=F(y). Letf₁(x₁e₁, y) be the homogeneous codimension-1 CK extension satisfyingf₁(0, y)=A₀(y)=f(0, y).Lemma 4 then asserts that for any ε>0,x₁e₁∈R¹,y∈R^q,

|f₁(x₁e₁, y)| ≤C_εe^(Ω+ε)|x1|≤C_εe^{(Ω+ε)|x1e1+y|}.

By invoking Theorem 1, we get supp(F)⊂B(0,Ω+ε). Letting ε!0, we conclude that supp(F)⊂B(0,Ω). The proof is complete.

The following variation of Theorem 2 holds.

Theorem 3. LetF be analytic, defined inR^q, taking values inC^(q), the com- plex Clifford algebra generated bye_p+1, ...,e_p+q, andF∈L²(R^q). LetΩbe a positive real number. Then the following assertions hold:

(1)If supp(F)⊂B(0,Ω), thenF has a homogeneous codimension-pCK exten- sion toR^p+q, denoted byf, and there exists a constant C such that

|f(x, y)| ≤Ce^Ω|x+y|, x∈R^p, y∈R^q.

(2) If there exists a homogeneous codimension-pCK extension of F toR^p+q, denoted byf and a constant C such that

|f(x, y)| ≤Ce^Ω|x+y|, x∈R^p, y∈R^q, thensupp(F)⊂B(0,√

2Ω).

Moreover, in each of the above cases, we have

f(x, y) = 1 (2π)^q

R^q

ε^p₁(x, y, t)F(t) dt, x∈R^p, y∈R^q.

Proof. The assertion (1) is the easy part. We only indicate how to prove (2).

Note that, if in Lemma 4, instead of the assumption

|f(x, y)| ≤Ce^Ω|x|

one adopts the weaker assumption

|f(x, y)| ≤Ce^Ω|x+y|, x∈R^p, y∈R^q, then the inequality (8) becomes

|∂_y^lf(0, y)| ≤C(m−1)m...(m+l−2) ρ^m+l−1 e

√2Ω|ρ⁺y|.

This will modify the inequality (9). By taking ρ=|x₁|(1+ε/Ω) in the modified inequality, we have, for anyε>0,

|f(x₁e₁, y)| ≤C_εe^{√2(Ω+ε)|x1e1+y|}, x₁e₁∈R¹, y∈R^q. The proof is complete by invoking Theorem 1.

Next we extend Theorem 2 to the generalized CK extension case involving a k-monogenic weight function.

Lemma 5. Let P_k(x)∈M⁺(p, k,C^(p)) be given. Denote byf_Pk(x, y) the gen- eralized CK extension of F(y) in relation to P_k, which is left-monogenic in R^m. Assume that

|f_Pk(x, y)| ≤Ce^Ω|x|, x∈R^p, y∈R^q,

whereΩis a positive real number. Letf₁(x₁e₁, y)be the homogeneous codimension- 1 CK extension of the same initial valuef₁(0, y)=F(y). Then for any ε>0, there exists a constant C_ε>0 such that

|f₁(x₁e₁, y)| ≤C_εe^(Ω+ε)|x1| for any x₁e₁∈R¹ andy∈R^q.

Proof. Sincef_Pk(x, y) is the generalized CK extension ofF(y) in relation toP_k, we have

f_Pk(x, y) = ∞ l=0

x^lP_k(x)F_l(y),

where

F_2l(y) =

(−1)^lΓ(k+p/2)∂_y^2lF(y) 2^2ll!Γ(l+k+p/2) , (11)

and

F_2l+1(y) =

(−1)^lΓ(k+p/2)∂_y^2l+1F(y) 2^2l+1l!Γ(l+k+p/2+1) . (12)

On the other hand, sincef_Pk is left-monogenic inR^m=R^p⊕R^q, by Cauchy’s inte- gral formula, we have

f_Pk(x, y) = 1 ω_m−1

∂B(x⁺y,ρ)

E(t−(x+y))dσ(t)f_Pk(t), whereρis any positive real number.

Then

∂^l+k

∂x^l₁^+kf_Pk(x, y) = 1 ω_m−1

∂^l+k

∂x^l₁^+k

∂B(x+y,ρ)

E(t−(x+y))dσ(t)f_Pk(t).

By Lemma 3 and the assumption|f_Pk(x, y)|≤Ce^Ω|x|, we have ∂^l+k

∂x^l₁^+kf_Pk(x, y)

≤C(m−1)m...(m+l+k−2)

ρ^l+k e^Ω(|x|+ρ). Therefore,

|F_l(y)|= lim

|x|!0

1 l!

∂^l+k

∂x^l₁^+kf_Pk(x, y) ≤C1

l!

(m−1)m...(m+l+k−2) ρ^l+k e^Ωρ. (13)

Using (11), (12) and the above estimate (13), we have

|f₁(x₁e₁, y)| ≤^∞

l=0

|x₁|^l|∂_y^l[F(y)]| l!

≤^∞

l=0

l+k+p 2

_k+p/2|x₁|^l(m−1)m...(m+l+k−2) l!ρ^l+k e^Ωρ.

For any ε>0, set l₀=[2Ω(m+k−2)/ε]+1, then l>l₀ implies (m+k−2)/l<ε/2Ω.

Thus, ifl>l₀, then 1+(m+k−2)/l<1+ε/2Ω, so

|f₁(x₁e₁, y)| ≤C ∞ l=0

l+k+p 2

_k+p/2|x₁|^l(m−1)m...(m+l+k−2) l!ρ^l+k e^Ωρ

≤C

l0

l=0

l+k+p 2

_k+p/2|x₁|^l(m−1)m...(m+l+k−2) l!ρ^l+k e^Ωρ +C_ε

∞ l=l0

l+k+p 2

_k+p/2|x₁|^l(1+ε/2Ω)^l−l0 ρ^l+k e^Ωρ, whereC_ε=(m−1)m...(m+k+l₀−2)/l₀!.

When|x₁|>1, takingρ=|x₁|(1+ε/Ω)², we have

|f₁(x₁e₁, y)| ≤C_εe^(Ω+ε)|x1|. (14)

When|x₁|<1, takingρ=2, we get thatf_Pk is bounded. So the inequality (14) also holds. The proof is complete.

Theorem 4. (Generalized codimension-p Paley–Wiener Theorem) Let P_k∈ M⁺(p, k;C^(p))be given,Fbe analytic, defined inR^q, taking values inC^(q), the com- plex Clifford algebra generated bye_p+1, ...,e_p+q, andF∈L²(R^q). LetΩbe a positive real number. Then the following two assertions are equivalent:

(1) F has a homogeneous codimension-p generalized CK extension to R^p+q, denoted byf_Pk, and there exists a constantC such that

|f_Pk(x, y)| ≤Ce^Ω|x| for any x∈R^p andy∈R^q. (2) supp(F)⊂B(0,Ω).

Moreover, if one of the above conditions holds, then we have

f_Pk(x, y) = 1 (2π)^q

R^q

ε^p_P

k(x, y, t)F(t)dt for any x∈R^p andy∈R^q. Proof. (2)⇒(1) Set

G_Pk(x, y) = 1 (2π)^q

R^q

ε^p_P

k(x, y, t)F(t)dt.

Using the same method as in the first part of Theorem 2, we can easily prove this implication.

(1)⇒(2) Let A₀(y)=F(y). By CK extension, using A₀(y), we can construct another left-monogenic functionf₁(x₁e₁, y) which satisfiesf₁(0, y)=A₀(y)=f(0, y).

If we can prove that for anyε>0,x₁e₁∈R¹,y∈R^q,

|f₁(x₁e₁, y)| ≤C_εe^{(Ω+ε)|x1e1+y|}, (15)

by Theorem 1, we get supp(F) ⊂B(0,Ω+ε). Lettingε!0, we get supp(F) ⊂B(0,Ω).

We are thus reduced to show inequality (15).

Since

f₁(x₁e₁, y) = ∞ l=0

(x₁e₁)^lA_l(y) = ∞

l=0

(−1)^l(x₁e₁)^l∂_y^l[A₀(y)]

l! ,

and

|f(x, y)| ≤Ce^Ω|x| for anyx∈R^p andy∈R^q,

by Lemma 5, for anyε>0, we have

|f₁(x₁e₁, y)| ≤C_εe^{(Ω+ε)|x1e1+y|} for anyx₁e₁∈R¹andy∈R^q. So inequality (15) holds. The proof is complete.

The rest of this section will deal with a PW theorem in relation to generalized Taylor series.

Lemma 6. Assume that f(x, y) is left-monogenic in R^m=R^p⊕R^q with the form(3) and such that

|f(x, y)| ≤Ce^Ω|x|, x∈R^p, y∈R^q, (16)

whereΩis a positive real number. Then for all k≥0 andα∈Ik, we have

|T_k,α(x, y)| ≤Ce^Ω|x|, x∈R^p, y∈R^q, whereC depends onk andα.

Proof. Based on (3) and using the orthogonality on the sphere betweenP_k(ω) andP_l(ω) (l=k), and that betweenP_k(ω) andωP_l(ω) for anykandl, we have

S^p−1

P_k,α(ω)f(|x|ω, y)dS_ω= ∞

l=0

(−1)^l|x|^2l+kT_k,α^(2l)(y).

According to the condition (16), we obtain ^∞

l=0

x^2lP_k(x)T_k,α^(2l)(y) ≤C

^∞

l=0

(−1)^l|x|^2l+kT_k,α^(2l)(y)

≤Ce^Ω|x|. Similarly, we have

S^p−1

ωP_k,α(ω)f(|x|ω, y)dS_ω= ∞

l=0

(−1)^l|x|^2l+1+kT_k,α^(2l+1)(y).

Using the condition (16), we also obtain ^∞

l=0

x^2l+1P_k(x)T_k,α^(2l+1)(y) ≤C

^∞

l=0

(−1)^l|x|^2l+1+kT_k,α^(2l+1)(y)

≤Ce^Ω|x|. Thus we have

|T_k,α(x, y)|= ^∞

l=0

x^lP_k,α(x)T_k,α^(l)(y)

≤ ^∞

l=0

x^2lP_k,α(x)T_k,α^(2l)(y) +

^∞

l=0

x^2l+1P_k,α(x)T_k,α^(2l+1)(y)

≤Ce^Ω|x|, x∈R^p, y∈R^q.