# On the Convolution Equation Related to the Diamond Klein-Gordon Operator

Authors: Amphon Liangprom; Kamsing Nonlaopon
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the distributioneαx(♢+m2)kδ for m≥0, where (♢+m2)k is the diamond Klein-Gordon operator iterated k times, δ is the Dirac delta
distribution, x=(x1,x2,…,xn) is a variable in ℝn, and α=(α1,α2,…,αn) is a
constant. In particular, we study the application of eαx(♢+m2)kδ for solving the solution of some convolution equation. We find that the types of solution of such convolution
equation, such as the ordinary function and the singular distribution, depend on the relationship between k and M.


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## Body

## 1. Introduction

Then-dimensional ultrahyperbolic operator □k iterated k times is defined by(1.1)□k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k,
where p+q=n is the dimension of ℝn, and k is a nonnegative integer. We consider the linear differential equation of the form(1.2)□ku(x)=f(x),
where u(x) and f(x) are generalized functions, and x=(x1,x2,…,xn)∈ℝn.Gelfand and Shilov [1] have first introduced the fundamental solution of (1.2), which was initially complicated. Later, Trione [2] has shown that the generalized function R2kH(x) defined by (2.2) with γ=2k is the unique fundamental solution of (1.2). Tellez [3] has also proved that R2kH(x) exists only when n=p+q with odd p.Kananthai [4] has first introduced the operator ♢k called the diamond operator iterated k times, which is defined by(1.3)♢k=[(∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2]k,
where n=p+q is the dimension of ℝn, for all x=(x1,x2,…,xn)∈ℝn and nonnegative integers k. The operator ♢k can be expressed in the form(1.4)♢k=▵k□k=□k▵k,
where □k is defined by (1.1), and ▵k is the Laplace operator iterated k times defined by(1.5)▵k=(∂2∂x12+∂2∂x22+⋯+∂2∂xn2)k.
Note that in case k=1, the generalized form of (1.5) is called the local fractional Laplace operator; see [5] for more details. On finding the fundamental solution of this product, he uses the convolution of functions which are fundamental solutions of the operators □k and ▵k. He found that the convolution (-1)kR2ke(x)*R2kH(x) is the fundamental solution of the operator ♢k, that is,(1.6)♢k((-1)kR2ke(x)*R2kH(x))=δ,
where R2kH(x) and R2ke(x) are defined by (2.2) and (2.7), respectively (with γ=2k), and δ is the Dirac-delta distribution. The fundamental solution (-1)kR2ke(x)*R2kH(x) is called the diamond kernel of Marcel Riesz. A number of effective results on the diamond kernel of Marcel Riesz have been presented by Kananthai [6–12].In 1997, Kananthai [13] has studied the properties of the distribution eαx□kδ and the application of the distribution eαx□kδ for finding the fundamental solution of the ultrahyperbolic equation by using the convolution method. Later in 1998, he has also studied the properties of the distribution eαx♢kδ and its application for solving the convolution equation(1.7)eαx♢kδ*u(x)=eαx∑r=0mCr♢rδ.
Recently, Nonlaopon gave some generalizations of his paper [6]; see [14] for more details.In 2000, Kananthai [15] has studied the application of the distribution eαx□kδ for solving the convolution equation(1.8)eαx□kδ*u(x)=eαx∑r=0mCr□rδ,
which is related to the ultrahyperbolic equation.In 2009, Sasopa and Nonlaopon [16] have studied the properties of the distribution eαx□ckδ and its application to solve the convolution equation(1.9)eαx□ckδ*u(x)=eαx∑r=0mCr□crδ.
Here, □ck is the operator related to the ultrahyperbolic type operator iterated k times, which is defined by(1.10)□ck=(1c2∑i=1p∂2∂xi2-∑j=p+1p+q∂2∂xj2)k,
where p+q=n is the dimension of ℝn.In 1988, Trione [17] has studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator iterated k times, which is defined by(1.11)(□+m2)k=(∑i=1p∂2∂xi2-∑j=p+1p+q∂2∂xj2+m2)k.
The fundamental solution of the operator (□+m2)k is given by(1.12)W2k(x,m)=∑r=0∞(-1)rΓ(k+r)r!Γ(k)(m2)r(-1)rR2k+2rH(x),
where R2k+2rH(x) is defined by (2.2) with γ=2k+2r. Next, Tellez [18] has studied the convolution product of Wα(x,m)*Wβ(x,m), where α and β are any complex parameter. In addition, Trione [19] has studied the fundamental (P±i0)λ-ultrahyperbolic solution of the Klein-Gordon operator iterated k times and the convolution of such fundamental solution.Liangprom and Nonlaopon [20] have studied the properties of the distribution eαx(□+m2)kδ and its application for solving the convolution equation(1.13)eαx(□+m2)kδ*u(x)=eαx∑r=0MCr(□+m2)rδ,
where (□+m2)k is defined by (1.11).In 2007, Tariboon and Kananthai [21] have introduced the operator (♢+m2)k called diamond Klein-Gordon operator iterated k times, which is defined by(1.14)(♢+m2)k=[(∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2+m2]k,
where p+q=n is the dimension of ℝn, for all x=(x1,x2,…,xn)∈ℝn,m≥0 and nonnegative integers k. Later, Lunnaree and Nonlaopon [22, 23] have studied the fundamental solution of operator (♢+m2)k, and this fundamental solution is called the diamond Klein-Gordon kernel. They have also studied the Fourier transform of the diamond Klein-Gordon kernel and its convolution.In this paper, we aim to study the properties of the distributioneαx(♢+m2)kδ and the application of eαx(♢+m2)kδ for solving the convolution equation(1.15)eαx(♢+m2)kδ*u(x)=eαx∑r=0MCr(♢+m2)rδ,
where (♢+m2)k is defined by (1.14), u(x) is the generalized function, and Cr is a constant. On finding the type of solution u(x) of (1.15), we use the method of convolution of the generalized functions.Before we proceed to that point, the following definitions and concepts require clarifications.


## 2. Preliminaries

Definition 2.1.
Letx=(x1,x2,…,xn) be a point of the n-dimensional Euclidean space ℝn. Let
(2.1)u=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2
be the nondegenerated quadratic form, where p+q=n is the dimension of ℝn. Let Γ+={x∈ℝn:x1>0andu>0} be the interior of a forward cone, and let Γ¯+ denote its closure. For any complex number γ, we define the function
(2.2)RγH(x)={u(γ-n)/2Kn(γ),forx∈Γ+,0,forx∉Γ+,
where the constant Kn(γ) is given by
(2.3)Kn(γ)=π(n-1)/2Γ((2+γ-n)/2)Γ((1-γ)/2)Γ(γ)Γ((2+γ-p)/2)Γ((p-γ)/2).The functionRγH(x) is called the ultrahyperbolic kernel of Marcel Riesz, which was introduced by Nozaki [24]. It is well known that RγH(x) is an ordinary function if Re(γ)≥n and is a distribution of γ if Re(γ)<n. Let supp RγH(x) denote the support of RγH(x) and suppose that supp RγH(x)⊂Γ¯+, that is, supp RγH(x) is compact.By puttingp=1 in R2kH(x) and taking into account Legendre's duplication formula(2.4)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
we obtain(2.5)IγH(x)=v(γ-n)/2Hn(γ),v=x12-x22-x32-⋯-xn2, where(2.6)Hn(γ)=π(n-2)/22γ-1Γ(γ+2-n2)Γ(γ2).
The function IγH(x) is called the hyperbolic kernel of Marcel Riesz.Definition 2.2.
Letx=(x1,x2,…,xn) be a point of ℝn and ω=x12+x22+⋯+xn2. The elliptic kernel of Marcel Riesz is defined by
(2.7)Rγe(x)=ω(γ-n)/2Wn(γ),
where n is the dimension of ℝn, γ∈ℂ, and
(2.8)Wn(γ)=πn/22γΓ(γ/2)Γ((n-γ)/2).Note thatn=p+q. By putting q=0 (i.e., n=p) in (2.2) and (2.3), we can reduce u(γ-n)/2 to ωp(γ-p)/2, where ωp=x12+x22+⋯+xp2, and reduce Kn(γ) to
(2.9)Kp(γ)=π(p-1)/2Γ((1-γ)/2)Γ(γ)Γ((p-γ)/2).Using the Legendre’s duplication formula(2.10)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),(2.11)Γ(12+z)Γ(12-z)=πsec(πz),
we obtain
(2.12)Kp(γ)=12sec(γπ2)Wp(γ).
Thus, in case q=0, we have
(2.13)RγH(x)=u(γ-p)/2Kp(γ)=2cos⁡(γπ2)u(γ-p)/2Wp(γ)=2cos⁡(γπ2)Rγe(x).
In addition, if γ=2k for some nonnegative integer k, then
(2.14)R2kH(x)=2(-1)kR2ke(x).Lemma 2.3.
The convolution(-1)kR2ke(x)*R2kH(x) is the fundamental solution of the diamond operator iterated k times, that is,
(2.15)♢k((-1)kR2ke(x)*R2kH(x))=δ.For the proof of this Lemma, see [4, 12].It can be shown thatR-2ke(x)*R-2kH(x)=(-1)k♢kδ, for all nonnegative integers k.Definition 2.4.
Letx=(x1,x2,…,xn) be a point of ℝn. The function Tγ(x,m) is defined by
(2.16)Tγ(x,m)=∑r=0∞(-γ2r)(m2)r(-1)γ/2+rRγ+2re(x)*Rγ+2rH(x),
where γ is a complex parameter, and m is a nonnegative real number. Here, Rγ+2rH(x) and Rγ+2re(x) are defined by (2.2) and (2.7), respectively.From the definition ofTγ(x,m), by putting γ=-2k, we have (2.17)T-2k(x,m)=∑r=0∞(kr)(m2)r(-1)-k+rR2(-k+r)e(x)*R2(-k+r)H(x).
Since the operator (♢+m2)k defined by (1.14) is linearly continuous and has 1-1 mapping, this possesses its own inverses. From Lemma 2.3, we obtain(2.18)T-2k(x,m)=∑r=0∞(kr)(m2)r♢k-rδ=(♢+m2)kδ.Substitutingk=0 in (2.18) yields that we have T0(x,m)=δ. On the other hand, putting γ=2k in (2.16) yields(2.19)T2k(x,m)=(-k0)(m2)0(-1)k+0R2k+0e(x)*R2k+0H(x)+∑r=1∞(-kr)(m2)r(-1)k+rR2k+2re(x)*R2k+2rH(x).
The second summand of the right-hand side of (2.19) vanishes when m=0. Hence, we obtain (2.20)T2k(x,m=0)=(-1)kR2ke(x)*R2kH(x),
which is the fundamental solution of the diamond operator.For the proofs of Lemmas2.5 and 2.6, see [23].Lemma 2.5.
Given the equation(2.21)(♢+m2)ku(x)=δ,
where (♢+m2)k is the diamond Klein-Gordon operator iterated k times, defined by
(2.22)(♢+m2)k=[(∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2+m2]k
with a nonnegative integer k and the Dirac-delta distribution δ, then u(x)=T2k(x,m) is the fundamental solution of the diamond Klein-Gordon operator iterated k times (♢+m2)k, where T2k(x,m) is defined by (2.16) with γ=2k.Lemma 2.6.
LetT2k(x,m) be the diamond Klein-Gordon kernel defined by (2.16), then T2k(x,m) is a tempered distribution and can be expressed by
(2.23)T2k(x,m)=T2k-2v(x,m)*T2v(x,m),
where v is a nonnegative integer and v<k. Moreover, if one puts l=k-v and h=v, then one obtains
(2.24)T2l(x,m)*T2h(x,m)=T2l+2h(x,m)
for l+h=k.


## 3. Properties of the Distributioneαx(♢+m2)kδ

Lemma 3.1.
The following equality holds:(3.1)eαx(♢+m2)kδ=Lkδ,
and eαx(♢+m2)kδ is the tempered distribution of order 4k with support {0}, where L is the partial differential operator and is defined by
(3.2)L≡(♢+m2)+∑r=1nαr2□-2∑r=1n∑i=1p(αr∂3∂xi2∂xr+αi∂3∂xi∂xr2)+2∑r=1n∑j=p+1p+q(αr∂3∂xj2∂xr+αj∂3∂xj∂xr2)+4∑r=1nαr(∑i=1pαi∂2∂xi∂xr-∑j=p+1p+qαj∂2∂xj∂xr)-2∑r=1nαr2(∑i=1pαi∂∂xi-∑j=p+1p+qαj∂∂xj)+(∑i=1pαi2-∑j=p+1p+qαj2)▵-2(∑i=1pαi2-∑j=p+1p+qαj2)∑r=1nαr∂∂xr+(∑i=1pαi2-∑j=p+1p+qαj2)∑r=1nαr2.
As before, □ is the ultrahyperbolic operator defined by (1.1) (with k=1), and ▵ is the Laplace operator defined by
(3.3)▵=∂2∂x1+∂2∂x2+⋯+∂2∂xn.Proof.
Letφ∈𝒟 be the space of testing functions which are infinitely differentiable with compact supports, and let 𝒟′ be the space of distributions. Now,
(3.4)〈eαx(♢+m2)δ,φ(x)〉=〈δ,(♢+m2)eαxφ(x)〉,
for eαx(♢+m2)δ∈𝒟′. A direct computation shows that
(3.5)(♢+m2)eαxφ(x)=eαxTφ(x),
where T is the partial differential operator defined by
(3.6)T≡(♢+m2)+∑r=1nαr2□+2∑r=1n∑i=1p(αr∂3∂xi2∂xr+αi∂3∂xi∂xr2)-2∑r=1n∑j=p+1p+q(αr∂3∂xj2∂xr+αj∂3∂xj∂xr2)+4∑r=1nαr(∑i=1pαi∂2∂xi∂xr-∑j=p+1p+qαj∂2∂xj∂xr)+2∑r=1nαr2(∑i=1pαi∂∂xi-∑j=p+1p+qαj∂∂xj)+(∑i=1pαi2-∑j=p+1p+qαj2)▵+2(∑i=1pαi2-∑j=p+1p+qαj2)∑r=1nαr∂∂xr+(∑i=1pαi2-∑j=p+1p+qαj2)∑r=1nαr2.
Thus,
(3.7)〈δ,(♢+m2)eαxφ(x)〉=〈δ,eαxTφ(x)〉=Tφ(0).
Since〈eαx(♢+m2)kδ,φ(x)〉=〈(♢+m2)kδ,eαxφ(x)〉 for every φ(x)∈𝒟 and eαx(♢+m2)kδ∈𝒟′, we have
(3.8)〈(♢+m2)kδ,eαxφ(x)〉=〈(♢+m2)k-1δ,(♢+m2)eαxφ(x)〉=〈(♢+m2)k-1δ,eαxTφ(x)〉=〈(♢+m2)k-2δ,(♢+m2)eαxTφ(x)〉=〈(♢+m2)k-2δ,eαxT(Tφ(x))〉=〈(♢+m2)k-2δ,eαxT2φ(x)〉.
Repeating this process (♢+m2) with k-2 times, we finally obtain
(3.9)〈(♢+m2)k-2δ,eαxT2φ(x)〉=〈δ,eαxTkφ(x)〉=Tkφ(0),
where Tk is the operator of (3.6) iterated k times. Now,
(3.10)Tkφ(0)=〈δ,Tkφ(x)〉=〈Lδ,Tk-1φ(x)〉,
by the operator L in (3.2) and the derivative of distribution. Continuing this process, we obtain Tkφ(0)=〈Lkδ,φ(x)〉 or 〈eαx(♢+m2)kδ,φ(x)〉=〈Lkδ,φ(x)〉. By equality of distributions, we obtain (3.1) as required. Since δ and its partial derivatives have support {0} which is compact, hence, by Schwartz [25], Lkδ are tempered distributions and Lkδ has order 4k. It follows that eαx(♢+m2)kδ is a tempered distribution of order 4k with point support {0} by (3.1). This completes the proof.Lemma 3.2 (boundedness property).
Let𝒟 be the space of testing functions and 𝒟′ the space of distributions. For every φ∈𝒟 and eαx(♢+m2)kδ∈𝒟′,
(3.11)|〈eαx(♢+m2)kδ,φ(x)〉|≤M,
for some constant M.Proof.
Note that we have〈eαx(♢+m2)kδ,φ(x)〉=〈(♢+m2)kδ,eαxφ(x)〉 for every φ(x)∈𝒟 and eαx(♢+m2)kδ∈𝒟′. Hence,
(3.12)〈(♢+m2)kδ,eαxφ(x)〉=〈(♢+m2)k-1δ,(♢+m2)eαxφ(x)〉=〈(♢+m2)k-1δ,eαxTφ(x)〉,
where T is defined by (3.6). Continuing this process for k-1 times, we will obtain
(3.13)〈eαx(♢+m2)kδ,φ(x)〉=〈δ,eαxTkφ(x)〉=Tkφ(0).
Since φ∈𝒟, so φ(0) is bounded, and also Tkφ(0) is bounded. It then follows that
(3.14)|〈eαx(♢+m2)kδ,φ(x)〉|=Tkφ(0)≤M,
for some constant M.


## 4. The Application of Distributioneαx(♢+m2)kδ

Theorem 4.1.
LetL be the partial differential operator defined by (3.2), and consider the equation
(4.1)Lu(x)=δ,
where u(x) is any distribution in 𝒟′, then u(x)=eαxT2(x,m) is the fundamental solution of the operator L, where T2(x,m) is defined by (2.16) with γ=2.Proof.
From (3.1) and (4.1), we can write eαx(♢+m2)δ*u(x)=Lu(x)=δ. Convolving both sides by eαxT2(x,m), we have
(4.2)eαxT2(x,m)*eαx(♢+m2)δ*u(x)=eαxT2(x,m)*δ,
then
(4.3)eαx(T2(x,m)*(♢+m2)δ)*u(x)=eαxT2(x,m),
or equivalently,
(4.4)eαx((♢+m2)T2(x,m))*u(x)=eαxT2(x,m).
Since (♢+m2)T2(x,m)=δ by Lemma 2.5 with k=1, we obtain
(4.5)(eαxδ)*u(x)=eαxT2(x,m).
Moreover, since eαxδ=δ, we have δ*u(x)=eαxT2(x,m). It then follows that u(x)=eαxT2(x,m) is the fundamental solution of the operator L.Theorem 4.2 (the generalization of Theorem4.1).
From Lemma3.1, consider that
(4.6)eαx(♢+m2)kδ*u(x)=δ,
or
(4.7)Lku(x)=δ,
then u(x)=eαxT2k(x,m) is the fundamental solution of the operator Lk.Proof.
We can prove it by using either (4.6) or (4.7). If we start with (4.6), by convolving both sides by eαxT2k(x,m), we obtain
(4.8)eαxT2k(x,m)*(eαx(♢+m2)kδ*u(x))=eαxT2k(x,m)*δ,
or eαx((♢+m2)kT2k(x,m))*u(x)=eαxT2k(x,m). Since (♢+m2)kT2k(x,m)=δ by Lemma 2.5, we have (eαxδ)*u(x)=eαxT2k(x,m) or u(x)=eαxT2k(x,m) as required.
If we use (4.7), by convolving both sides by eαxT2(x,m), we obtain
(4.9)eαxT2(x,m)*Lku(x)=eαxT2(x,m)*δ,
or L(eαxT2(x,m))*Lk-1u(x)=eαxT2(x,m). By Theorem 4.1, we obtain Lk-1u(x)=eαxT2(x,m). Keeping on convolving eαxT2(x,m) for k-1 times, we finally obtain
(4.10)u(x)=eαx(T2(x,m)*T2(x,m)*⋯*T2(x,m))=eαxT2k(x,m),
by Lemma 2.6 and [26, page 196].In particular, if we putα=(α1,α2,…,αn)=0 in (4.6), then (4.6) reduces to (2.21), and we obtain u(x)=T2k(x,m) as the fundamental solution of the diamond Klein-Gordon operator iterated k times.Theorem 4.3.
Given the convolution equation(4.11)eαx(♢+m2)kδ*u(x)=eαx∑r=0MCr(♢+m2)rδ,
where (♢+m2)k is the diamond Klein-Gordon operator iterated k times defined by
(4.12)(♢+m2)k=(∑i=1p∂2∂xi2-∑j=p+1p+q∂2∂xj2+m2)k,
the variable x=(x1,x2,…,xn)∈ℝn, the constant α=(α1,α2,…,αn)∈ℝn, m is a nonnegative real number, δ is the Dirac-delta distribution with (♢+m2)0δ=δ,(♢+m2)1δ=(♢+m2)δ, and Cr is a constant, then the type of solution u(x) of (4.11) depends on k,M, and α as follows:  (1)
ifM<k and M=0, then the solution of (4.11) is
(4.13)u(x)=C0eαxT2k(x,m),
where T2k(x,m) is defined by (2.16) with γ=2k. If 2k≥n and for any α, then eαxT2k(x,m) is the ordinary function,(2)
if0<M<k, then the solution of (4.11) is
(4.14)u(x)=eαx∑r=1MCrT2k-2r(x,m),
which is an ordinary function for 2k-2r≥n with any arbitrary constant α,(3)
ifM≥k and for any α one supposes that k≤M≤N, then (4.11) has
(4.15)u(x)=eαx∑r=kNCr(♢+m2)r-kδ
as a solution which is the singular distribution.Proof.
(1)
ForM<k and M=0, then (4.11) becomes(4.16)eαx(♢+m2)kδ*u(x)=C0eαxδ=C0δ,
and by Theorem 4.2, we obtain
(4.17)u(x)=C0eαxT2k(x,m).
Now, by (2.2) and (2.7), R2ke(x) and R2kH(x) are ordinary functions, respectively, for 2k≥n. It then follows that C0eαxT2k(x,m) is an ordinary function for 2k≥n with any α.(2)
For0<M<k, then we can write (4.11) as(4.18)eαx(♢+m2)kδ*u(x)=eαx[C1(♢+m2)δ+C2(♢+m2)2δ+⋯+CM(♢+m2)Mδ].
Convolving both sides by eαxT2k(x,m) and applying Lemma 2.5, we obtain
(4.19)u(x)=eαx[C1(♢+m2)T2k(x,m)+C2(♢+m2)2T2k(x,m)+⋯+CM(♢+m2)MT2k(x,m)].
It is known that (♢+m2)kT2k(x,m)=δ, thus (♢+m2)k-r(♢+m2)rT2k(x,m)=δ for r<k. Convolving both sides by T2k-2r(x,m), we obtain
(4.20)T2k-2r(x,m)*(♢+m2)k-r(♢+m2)rT2k(x,m)=T2k-2r(x,m),
or
(4.21)(♢+m2)k-rT2k-2r(x,m)*(♢+m2)rT2k(x,m)=T2k-2r(x,m),
which leads to
(4.22)(♢+m2)rT2k(x,m)=T2k-2r(x,m),
for r<k. It follows that
(4.23)u(x)=eαx[C1T2k-2(x,m)+C2T2k-4(x,m)+⋯+CMT2k-2M(x,m)],
or
(4.24)u(x)=eαx∑r=1MCrT2k-2r(x,m).
Similarly, by case (1), eαxT2k-2r(x,m) is the ordinary function for 2k-2r≥n with any α. It follows that
(4.25)u(x)=eαx∑r=1MCrT2k-2r(x,m)
is also the ordinary function with any α.(3)
ifM≥k and for any α, we suppose that k≤M≤N, then (4.11) becomes(4.26)eαx(♢+m2)kδ*u(x)=eαx[Ck(♢+m2)kδ+Ck+1(♢+m2)k+1δ+⋯+CN(♢+m2)Nδ].
Convolving both sides by eαxT2k(x,m) and applying Lemma 2.5, we have
(4.27)u(x)=eαx[Ck(♢+m2)kT2k(x,m)+Ck+1(♢+m2)k+1T2k(x,m)+⋯+CN(♢+m2)NT2k(x,m)].
Now,
(4.28)(♢+m2)MT2k(x,m)=(♢+m2)M-k(♢+m2)kT2k(x,m)=(♢+m2)M-k,
for k≤M≤N. Thus,
(4.29)u(x)=eαx[Ckδ+Ck+1(♢+m2)δ+Ck+2(♢+m2)2δ+⋯+CN(♢+m2)N-kδ]=eαx∑r=kNCr(♢+m2)r-kδ.
Now, by (3.1) and (3.2), we have
(4.30)eαx(♢+m2)r-kδ=(♢+m2)r-kδ+(termsoflowerorderofpartialderivativeofδ)
for k≤r≤N. Since all terms on the right-hand side of this equation are singular distribution, it follows that
(4.31)u(x)=eαx∑r=kNCr(♢+m2)r-kδ
is the singular distribution. This completes the proof.



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*Source: 908491-2011-10-26.xml*