Applied Mathematics E-Notes, 1(2001), 34-39 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Two Special Convolution Products of (n/2 − k − 1)-th Derivatives of Dirac Delta in Hypercone ^{∗}

Manuel A. Aguirre T.^{†}

Received 25 October 2000

Abstract In this paper two special convolution products δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u) and δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u) are expressed in terms of several known quantities.

Let x = (x_{1,}x_{2}, ...x_{n}) be a point of R^{n}. We shall write

x_{1}^{2} + ... + x_{μ}^{2} − x_{μ+1}^{2} − ...x_{μ+ν}^{2} = u,  (1)

where μ + ν = n. Γ_{+} denotes the interior of the forward cone

Γ_{+} = {x ∈ R^{n}| x_{1} > 0, u > 0} ,  (2)

and Γ̄_{+} denotes its closure. Similarly, Γ_{−} denotes the domain

Γ_{−} = {x ∈ R^{n}| x_{1} < 0, u > 0}  (3)

and Γ̄_{−} denotes its closure. Let F (λ) be a function of the scalar variable λ, and let Φ = Φ(x) be a function endowed with the following properties: (i) Φ(x) = F (u), (ii) suppΦ(x) ⊂ Γ̄_{+} and (iii) e^{⟨x,y⟩}Φ(x) ∈ L_{1} if y ∈ V_{−}, where

V_{−} = {y ∈ R^{n}| y_{1} > 0, y_{1}^{2} + ... + y_{μ}^{2} − y_{μ+1}^{2} − ...y_{μ+ν}^{2} > 0} .  (4)

We let R denote the family of functions Φ.

Similarly, A denotes the family of functions of the form Φ = Φ(x) which satisfies conditions: (a) Φ(x) = F (u), (b) suppΦ(x) ⊂ Γ̄_{−}, and (c) e^{⟨x,y⟩}Φ(x) ∈ L_{1} if y ∈ V_{+}, where

V_{+} = {y ∈ R^{n}| y_{1} < 0, y_{1}^{2} + ... + y_{μ}^{2} − y_{μ+1}^{2} − ...y_{μ+ν}^{2} > 0} .  (5)

We shall consider the following functions of the family R_{α} introduced in [1, p.72]: R_{α}(u) = 0 if x ∉ Γ_{+}, and

R_{α}(u) = \frac{1}{K_{n}(α)}u^{(α−n)/2}, x ∈ Γ_{+}.

∗Mathematics Subject Classifications: 46F10, 46F12.

†(Núcleo Consolidado de Matemática Pura y Aplicada) Facultad de Ciencias Exactas, Universidad Nacional del Centro Pinto, 399, (7000) Tandil, Argentina, email:maguirre@exa.unicen.edu.ar

Here α is a complex parameter and n the dimension of the space, the constant K_{n}(α) is defined by

K_{n}(α) = \frac{π^{(n−1)/2}Γ((2 + α − n)/2)Γ((1 − α)/2))Γ(α)}{Γ((2 + α − μ)/2)Γ((μ − α)/2)}  (6)

and μ is the number of positive terms of

u = x_{1}^{2} + ... + x_{μ}^{2} − x_{μ+1}^{2} − ... − x_{μ+ν}^{2}, μ + ν = n.  (7)

R_{α}(u) is a distribution of α and is an ordinary function if the real part of α is greater than or equal to n.

By putting μ = 1 in R_{α}(u) and (6) and remembering the Legendre’s duplication formula of Γ(z) [2, p.344]

Γ(2z) = 2^{2z−1}π^{−1/2}Γ(z)Γ(z + 1/2),  (8)

R_{α}(u) reduces to M_{α}(u) which is the hyperbolic kernel of Riesz [3, p.31]: M_{α}(u) = 0 if x ∉ Γ_{+} and

M_{α}(u) = \frac{1}{H_{n}(α)}u^{(α−n)/2}, x ∈ Γ_{+}.  (9)

Here

u = x_{1}^{2} − x_{2}^{2} − ...x_{n}^{2},  (10)

and

H_{n}(α) = π^{(n−2)/2}2^{α−1}Γ(α/2)Γ((α − n + 2)/2).  (11)

Trione in [4, p.11] proves the validity of the property

♢^{k}R_{2k}(u) = R_{0}(u) = δ(x)  (12)

for k = 0, 1, 2, ..., where

♢^{k} = \{\frac{∂^{2}}{∂x_{1}^{2}} + ...\frac{∂^{2}}{∂x_{μ}^{2}} − \frac{∂^{2}}{∂x_{μ+1}^{2}} − ...\frac{∂^{2}}{∂x_{μ+ν}^{2}} \}^{k}  (13)

is the ultrahyperbolic operator iterated k-times and δ(x) = δ(x_{1}, x_{2}, ...x_{n}) is the Dirac delta function. From (12), R_{2k}(u) is the unique elementary solution of the n-dimensional ultrahyperbolic operator iterated k-times defined by (13)

Aguirre in [5, p.149] proves the following properties:

1. μ odd and ν even (n odd)

R_{2k}(u) = \frac{1}{(−1)^{(μ−1)/2}π^{(n−1)/2}2^{2k−1}Γ(k)} · \frac{u^{k−n/2}}{Γ(k − n/2 + 1)},  (14)

where u is defined by (7)

2. μ odd and ν odd (n even)

R_{2k}(u) = \frac{1}{(−1)^{(μ−1)/2}π^{(n−1)/2}2^{2k−1}Γ(k)} · \frac{u^{k−n/2}}{Γ(k − n/2 + 1)}  (15)

if k ≥ n/2 and

R_{2k}(u) = \frac{1}{(−1)^{(μ−1)/2}π^{(n−1)/2}2^{2k−1}Γ(k)}δ^{(n/2−k−1)}  (16)

if k < n/2.

On the other hand Aguirre and Trione in [6, p.123] prove the following formula

R_{α}(u) ∗ R_{2k}(u) = R_{α+2k}(u)  (17)

for all μ and ν where the symbol ∗ stands for convolution and μ+ν = n is the dimension of the space.

In this paper we find two formulae for two special convolution products

δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u)

and

δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u)

using the formulae (16), (17) and the following formula [10, p.123]

δ^{(k−1)}(m^{2} + u) = ∑_{ν=o}^{n/2−k−1} \frac{(m^{2})^{ν}}{ν!} δ^{(k+ν−1)}(u)  (18)

which holds when n is even and k < n/2 − 1.

THEOREM 1. Let k and l be non-negative integers and n an even positive integer. Then

δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u) = A_{k,l,n}δ^{(n/2−k−l−1)}(u)  (19)

under conditions (i) μ and ν are both odd, and (ii) 0 ≤ k +l ≤ n/2−1, where [5, p.148]

A_{k,l,n} = \frac{1}{2}(−1)^{(μ−1)/2}π^{(n−2)/2} \frac{Γ(k)Γ(l)}{Γ(k + l)},  (20)

and

δ^{(n/2−k−1)}(u) = \frac{(n/2 − k − 1)!}{(−1)^{n/2−k−1}}res_{β=−(n/2−k)}u^{β}  (21)

and u is defined by (7)

PROOF. From (16) and (17), we have,

δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u)
= (−1)^{(μ−1)/2}π^{(n−2)/2}2^{2k−1}Γ(k)
×Γ(l)(−1)^{(μ−1)/2}π^{(n−2)/2}2^{2l−1} (R_{2k}(u) ∗ R_{2l}(u))
= Γ(k)Γ(l)(−1)^{μ−1}π^{n−2}2^{2(k+l)−2}R_{2(k+l)}(u)  (22)

if μ and ν are both odd. Now from (22) and (16), we have

δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u) = \frac{1}{2}(−1)^{(μ−1)/2}π^{(n−2)/2} \frac{Γ(k)Γ(l)}{Γ(k + l)}δ^{(n/2−k−l−1)}(u) (23)

if μ and ν are both odd and k + l < n/2. From (23) we deduce (19). The proof is complete.

We remark that our new formula (19) is not a consequence of the convolution product

δ^{(s)}(P_{+}) ∗ δ^{(t)}(P_{+}) = (−1)^{s+t}h_{s,t,n}P_{−}^{n/2−s−t−2}  (24)

which appear in [7], where

h_{s,t,n} = 2^{−2}(−1)^{(ν−1)/2}π^{(n+2)/2} \frac{Γ(n/2 − s − 1)Γ(n/2 − t − 1)}{Γ(n/2 − s − t − 1)Γ(n − s − t − 2)}  (25)

and [11, p.278]

δ^{(s)}(P_{+}) = (−1)^{s}s!res_{λ=−s−1,s=0,1,2,...}P_{+}^{λ}  (26)

and [11, p.254]

⟨P_{+}^{λ}, φ⟩ = ∫_{P >0}(P (x))^{λ}φ(x)dx  (27)

and P (x) = u(x) = u with u defined by (7).

In fact, putting s = n/2 − k − 1 and t = n/2 − l − 1 in (24) and (25), we have

δ^{(n/2−k−1)}(P_{+}) ∗ δ^{(n/2−l−1)}(P_{+})
= 2^{−2}(−1)^{(μ−1)/2}π^{(n−2)/2} \frac{Γ(k)Γ(l)(−1)^{n−k−l}}{Γ(k + l)Γ(k + l − n/2 + 1)}P_{−}^{k+l−n/2}  (28)

provided n/2 ≤ k + l ≤ n − 2, where [11, p.269]

⟨P_{−}^{λ}, φ⟩ = ∫_{P <0}(−P (x))^{λ}φ(x)dx.  (29)

Therefore our new convolution product formula (19) is in some way complementary to the formula (28).

THEOREM 2. Let k and l be non-negative integers and n an even positive integer. Then

δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u)

= δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u) + a_{μ,n} ∑_{ν=1}^{k−1} \frac{(m^{2})^{ν}}{ν!} A_{k−ν,l}δ^{(n/2−k−l+ν−1)}(u)

+a_{μ,n} ∑_{j=1}^{l−1} \frac{(m^{2})^{j}}{j!} A_{k,l−j}δ^{(n/2−k−l+j−1)}(u) +

+a_{μ,n} ∑_{ν=1}^{k−1} ∑_{j=1}^{l−1} \frac{(m^{2})^{ν+j}}{ν!} A_{k−ν,l−j}δ^{(n/2−k−l+ν+j−1)}(u)  (30)

under conditions a) μ and ν are both odd, and b) 0 ≤ k + l ≤ n/2 − 1, where

A_{s,t} = \frac{Γ(s)Γ(t)}{Γ(s + t)}  (31)

38 Convolution Products and

a_{μ,n} = \frac{1}{2}(−1)^{(μ−1)/2}π^{(n−1)/2}.  (32)

PROOF. From (18), we have

δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u)

= ∑_{ν=0}^{k−1} ∑_{j=0}^{l−1} \frac{(m^{2})^{ν+j}}{ν!} \{A_{k−ν,l−j}δ^{(n/2−k−l+ν+j−1)}(u)\} .  (33)

Now using the formula (19), we have

δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u)

= ∑_{ν=0}^{k−1} ∑_{j=0}^{l−1} \frac{(m^{2})^{ν+j}}{ν!} \{A_{k−ν,l−j,n}δ^{(n/2−k−l+ν+j−1)}(u)\} .  (34)

where A_{k−ν,l−j,n} is defined by (20). From (34) and (19), we obtain

δ^{(n/2−k−1)}(m^{2} + u) ∗ δ^{(n/2−l−1)}(m^{2} + u)

= δ^{(n/2−k−1)}(u) ∗ δ^{(n/2−l−1)}(u) + +a_{μ,n} ∑_{ν=1}^{k−1} \frac{(m^{2})^{ν}}{ν!} A_{k−ν,l}δ^{(n/2−k−l+ν−1)}(u)

+a_{μ,n} ∑_{j=1}^{l−1} \frac{(m^{2})^{j}}{j!} A_{k,l−j}δ^{(n/2−k−l+j−1)}(u)

a_{μ,n} ∑_{ν=1}^{k−1} ∑_{j=1}^{l−1} \frac{(m^{2})^{ν+j}}{ν!j!} A_{k−ν,l−j}δ^{(n/2−k−l+ν+j−1)}(u)  (35)

where a_{μ,n} is defined by the (32). The formula (35) coincides with (30).

It’s clear that putting m^{2} = 0 in (30) we obtain (19).

We remark that (30) is not a consequence of the equality

δ^{(s−1)}(m^{2} + P) ∗ δ^{(t−1)}(m^{2} + P)

= δ^{(s−1)}(P_{+}) ∗ δ^{(t−1)}(P_{+}) + ∑_{r=1}^{n/2−s−t} \frac{(m^{2})^{r}}{r!} C_{n/2−s−r,n/2−t−r,r,l}P_{−}^{n/2−k−t−r}(36)

which appear in [8], where

C_{n/2−s−r,n/2−t−r,r,l}

= π^{n/2+1}2^{−2}(−1)^{(ν−1)/2} \frac{Γ(n/2 − s − r)Γ(n/2 − t − r)}{Γ(n/2 − s − t + 1)Γ(n − s − r − t − r)}.  (37)

In fact, putting s = n/2 − k − 1 and t = n/2 − l − 1 in (36) and (37), we have

δ^{(n/2−k−1)}(m^{2} + P) ∗ δ^{(n/2−l−1)}(m^{2} + P)

= δ^{(n/2−k−1)}(P_{+}) ∗ δ^{(n/2−l−1)}(P_{+}) + ∑_{r=1}^{k+l−n/2} \frac{(m^{2})^{r}}{r!} C_{k−r,l−r,r,l}P_{−}^{k+l−n/2−r}

under conditions μ and ν odd and k + l ≥ n/2 + r, where

C_{k−r,l−r,r,l} = π^{n/2+1}2^{−2}(−1)^{(ν−1)/2} \frac{Γ(k − r)Γ(l − r)}{Γ(k + l − n/2)Γ(k + l − 2r)}.

Therefore (30) is in some way complementary to (36).

References

[1] Y. Nozaki, On Riemann-Liouville integral of ultra-hyperbolic type, Kodai Mathematical Seminar Reports, 6(2)(1964), 69-87.

[2] A. Erdelyi, Ed. Higher Trascendental Functions, Vol. I, McGraw-Hill, New York, 1953.

[3] M. Riesz, L’intégrale de Riemann-Liouville et le probléme de Cauchy pour l’équation des ondes, Comm. Sém. Math. Univ. de Lund, 4(1939), 28-42.

[4] S. E. Trione, On Marcel Riesz’s ultrahyperbolic kernel, preprint.

[5] M. A. Aguirre T., The distributional Hankel transform of Marcel Riesz’s ultrahyperbolic kernel, Studies in Applied Mathematics 93(1994), 133-162.

[6] M. A. Aguirre T. and S. E. Trione, The distributional convolution products of Marcel Riesz’s ultra-hyperbolic kernel, Revista de la Unión Matemática Argentina, 39(1995), 115-124.

[7] M. A. Aguirre T., The distributional convolution product of k-th derivative of Dirac delta in hypercone, preprint.

[8] M. A. Aguirre T., Convolution product of (k − 1)-th derivative of Dirac’s delta in m^{2} + P, preprint.

[9] M. A. Aguirre T., The expansion of δ^{(k−1)}(m^{2}+P), Integral Transform and Special Functions, 8(1-2)(1999), 139-148..

[10] M. A. Aguirre T., The expansion and Fourier’s transform of δ^{(k−1)}(m^{2} + P), Integral Transform and Special Functions, 3(2)(1995), 113-134.

[11] I. M.Gelfand and G. E. Shilov., Generalized Function, Vol. I, Academic Press, New York, 1964.