Revista de Matema´tica: Teor´ıa y Aplicaciones 4(2): 13–23 (1997)
the hankel transform of causal distributions
Manuel A. Aguirre1
Abstract
In this note we evaluate the unidimensional distributional Hankel transform of
xα−1+
Γ(α)
,
xα−1
−
Γ(α)
,
|x|α−1
Γ(α/2)
,
|x|α−1sgn (x)
Γ(
α+1
2
)
and (x ± i0)α−1 and then we extend the formu-
lae to certain kinds of n-dimensional distributions called “causal” and “anti-causal”
distributions. We evaluate the distributional Hankel transform of
(m2 + P )α−1+
Γ(α)
,
(m2 + P )α−1
−
Γ(α)
,
|m2 + P |α−1
Γ
(
α
2
) , |m2 + P |α−1sgn (m2 + P )
Γ
(
α+1
2
) and (m2 + P ± i0)α−1.
1 Introduction
Let U(t) ∈ S′
R+
, where S′
R+
is the dual of SR+ (SR+ is the space of functions f ∈ S defined
in R+ = {t : t > 0}).
The Hankel transform of U(t) will be, by definition, the distribution V (S) ∈ S′
R+
defined by the formula
〈H{U(t)}, ϕ(s)〉 = 〈U(t),H{ϕ(s)}〉, (1)
for every ϕ ∈ SR+ (cf. [6], page 26, formula (I,5,6)).
By the Hankel transform of the function f(x) we mean the function g(s), 0 ≤ s <∞,
defined by the formula
g(s) = (H{f(t)}) = 1
2
∫
∞
0
f(t)t
n−2
2 Rn−2
2
(
√
ts)dt , (2)
where
Rγ(x) =
Jγ(x)
xγ
, (3)
and
Jγ(x) =
∞∑
ν=0
(−1)ν(x2 )γ+2ν
ν ! Γ(γ + ν + 1)
. (4)
1Facultad de Ciencias Exactas de la Universidad Nacional del Centro de la Provincia de
Buenos Aires, Pinto 399, 3er. piso, 7000 Tandil, Argentina.
13
14 m. aguirre
Similarly, let U(t) ∈ S′
R−
. The Hankel transform of U(t) will be, by definition, the
distribution V (s) ∈ S′
R−
, defined by the formula:
〈H{U(t)}, ϕ(s)〉 = 〈U(t),H{ϕ(s)}〉 (5)
for every ϕ ∈ SR− . Here S′R− designates the dual of SR− .
By SR− we designate the space of functions f ∈ S defined in the negative half line
R− = {t : t < 0}. By the Hankel transform of the function ϕ(s) we mean the function
h(t), −∞ < t < 0, defined by the formula:
h(t) = H{ϕ(s)} = (−1)1
2
∫ 0
−∞
ϕ(s)S
n−2
2 (
√
st)ds (6)
Let φs be a distribution of variable s, and let U(x) ∈ C∞(Rn) be such that the
(n-1)-dimensional manifold U(x1, . . . , xn) = 0 has no critical points; ϕu(x) denotes the
distribution defined on Rn be the formula (called the Leray formula ([2], p. 102)
∫
Rn
φu(x)f(x)dx1 . . . dxn =
∫
∞
−∞
φsds
∫
u(x)=s
f(x)Wn(x, dx) ; (7)
here wn is an (n-1)-dimensional form on u defined as follows:
du ∧ dw = dx1 ∧ dx2 ∧ . . . ∧ dxn ;
the manifold u(x) = s has the orientation such that Wn(x, dx) > 0.
On the other hand, from [5], p. 2, formula (11) we have,
〈H{Φ(U(x))}, ϕ〉 = 〈H{V (t)}, ϕ(s)〉 (8)
where
V (t) = Φ(U) (9)
and
ψ(s) =
∫
U=S
ϕW (x, dx) (10)
In this note we shall evaluate the unidimensional distributional Hankel transform of
xα−1+
Γ(α)
,
xα−1
−
Γ(α)
,
|x|α−1
Γ(α2 )
,
|x|α−1sgn(x)
Γ(α+12 )
, and (x ± i0)α−1 (cf. formulae (32), (34), (36), (37),
(41) and (42) and then we extend the formulae to distributions called “causal” and “an-
ticausal” distribution. We evaluate the distributional Hankel transform of
(m2 + P )α−1+
Γ(α)
,
(m2 + P )α−1−
Γ(α)
,
(m2 + P )α−1
Γ(α2 )
,
(m2 + P )α−1sgn(m2 + P )
Γ(α+12 )
and (m2+P±i0)α−1 (cf. formulae
(72), (75), (78), (81) and (83)).
the hankel transform of causal distributions 15
2 The distributional Hankel transform of
x
α−1
+
Γ(α)
,
x
α−1
−
Γ(α)
,
x
α−1
Γ(α2 )
,
|x|α−1sgn(x)
Γ(α−12 )
, and (x± i0)α−1.
Consider the family of generalized functions defined by the formulae,
R+α (x) =
xα−1+
Γ(α)
, (11)
R−α (x) =
xα−1−
Γ(α)
, (12)
Yα(x) =
|x|α−1
Γ(α2 )
, (13)
and
Zα(x) =
|x|α−1sgn(x)
Γ(α−12 )
. (14)
Let fα(x± i0) be a function defined by
fα(x± i0) = (x± i0)α−1 = xα−1+ + e±(α−1)pii)xα−1− (15)
where,
xλ+ =
{
xλ for x > 0
0 for x ≤ 0 , (16)
xλ− =
{ |x|λ for x < 0
0 for x ≥ 0 , (17)
|x|λ = xλ+ + xλ− , (18)
|x|λsgn(x) = xλ+ − xλ− , (19)
(x± i0)λ = lim
ε→0
(x± iε)λ , (20)
λ is a complex number and
sgn(x) =
{
1 if x > 0
−1 if x < 0 . (21)
Gelfand and Shilov ([3], p. 48–60) have studied in detail the functions (16), (17), (18),
(19) and (20). These functions are locally integrables and define generalized functions for
all λ 6= −1,−2, . . .; except (20) which is an entire function of λ.
The distributions (16), (17), (18) and (19) are analytic in λ except at λ = −1,−2, . . .,
where they have simple poles, therefore from [3], pag. 57-58 and taking into account the
formula,
Resα=−kΓ(α) =
(−1)k
k !
, (22)
16 m. aguirre
we have
lim
α→−k
R+α (x) = δ
(k)(x) , (23)
lim
α→−k
R−α (x) = (−1)kδ(k)(x) , (24)
lim
α→−2k
Yα(x) =
(−1)kk !
(2k)!
δ(2k)(x) , (25)
and
lim
α→−1k−1
Zα(x) =
(−1)k+1k !
(2k + 1)!
δ(2k+1)(x) , (26)
where
Γ(α) =
∫
∞
0
e−xxα−1dx . (27)
By taking into account the formula,
∫
∞
0
xuJν(ax)dx =
2ua−u−1Γ(12 +
ν
2 +
µ
2 )
Γ(12 +
ν
2 +
µ
2 )
(28)
− Real ν − 1 < Real µ < 12 , a > 0 ([9], page 684, formula 14).
From the formulae (1) and (5) we shall get the Hankel distributional transform of the
family of generalized functions defined by the formulae (11), (12), (13), (14) and (15).
In fact, from (1) and (2), we have,
〈H{R+α (t)}, ϕ〉 =
〈
H
[
tα−1
+
Γ(α)
]
, ϕ
〉
=
=
〈 tα−1+
Γ(α)
,H[ϕ(s)]
〉
=
1
Γ(α)
∫
∞
0
tα−1H[ϕ(s)]dt =
=
〈1
2
S
n−2
2
1
Γ(α)
∫
∞
0
tα−1Rn−2
2
(
√
st)dt, ϕ(s)
〉
. (29)
On the other hand, taking into account the formulae (3) and (28) we have,
1
2
1
Γ(α)
∫
∞
0
tα−1Rn−2
2
(
√
st)dt =
=
1
Γ(α)
∫
∞
0
t2α−
n
2 S−
n
4
+ 1
2Jn−2
2
(
√
s . t)dt =
=
S−
n
4
+ 1
2
Γ(α)
[22α−n2 S−α+n4+ 12Γ(α)
Γ(n2 − α)
]
=
=
S−α
2
n
2
−2αΓ(n2 − α)
. (30)
Therefore, from (29) and taking into account the formulae (30) and (31) we have,
〈H{R+α (t)}, ϕ〉 =
1
2
n
2
−2α
〈R+n
2
−α
(s), ϕ〉 . (31)
the hankel transform of causal distributions 17
From (31) we obtain the following formula:
H{R+α (t)} =
1
2
n
2
−2α
R+n
2
−α
(s) (32)
Similarly, from (5) and (6) we have,
〈H{R−α (t)}, ϕ(s)〉 = 〈R−α (t),H{ϕ(s)}〉 =
=
∫ 0
−∞
|t|α−1
Γ(α)
H{ϕ(s)}dt =
=
1
Γ(α)
∫
∞
0
tα−1
{
(−1)1
2
∫ 0
−∞
ϕ(s)S
n−2
2 Rn−2
2
(
√
st)ds
}
dt =
=
1
Γ(α)
(−1)
2
〈
(−S)n−22
∫
∞
0
tα−1Rn−2
2
(
√
st)dt, ϕ(−s)
〉
. (33)
Taking into account the formula (30) and the formula (12), from (33) we have,
〈H{R−α (t)}, ϕ(s)〉 =
(−1)n2
2
n
2
−2α
〈S−α+n−22
Γ(n2 − α)
, ϕ(s)
〉
=
=
〈(−1)n2
2
n
2
−2α
R−n
2
−α
(s), ϕ(s)
〉
,
therefore,
H{R−α (t)} =
(−1)n2
2
n
2
−2α
R−n
2
−α
(s) (34)
On the other hand, the family of generalized functions defined by the formulae (13)
and (14) are even and odd combinations of R+α (t) and R
−
α (t), therefore to obtain its Hankel
transform we take into account the results (32) and (34). In fact, from (13), (18), (32),
(34) and the Lagrange’s duplication formula
Γ(2Z) = 22Z−1pi−
1
2Γ(Z)Γ(Z +
1
2
) (35)
([8], page 344, formula 15), we have
H{Yα(t)} = H
{ |t|α−1
Γ(α2 )
}
=
=
Γ(α+12 )
2
n
2
−3α+1pi
1
2
[R+n
2
−α
(s) + (−1)n2R−n
2
−α
(s)] (36)
Similary, from (14), (19), (32), (34) and taking into account the formula (35), we have,
H{Zα(t)} = H
{ |t|α−1 sgn(t)
Γ(α+12 )
}
=
=
Γ(α2 )
2
n
2
−3α+1pi
1
2
[R+n
2
−α
(s)− (−1)n2R−n
2
−α
(s)] . (37)
18 m. aguirre
Finally to obtain the Hankel transform of fα(x ± i0) we shall take into account the
results (32) and (34).
In fact, for n even, from (15) and taking into account the formulae (11), (12), (32) and
(34) we have,
H{fα(t+ i0)} = H{tα−1+ }+ e(α−1)piiH{tα−1− } = (38)
= Γ(α)[H{R+α (t)}+ e(α−1)piiH{R−α (t)}] = (39)
= Γ(α)
2
n
2
−2α
[
S
n
2
−α−1
+
Γ(n
2
−α)
+ eα−1)pii(−1)n2 S
n
2
−α−1
−
Γ(n
2
−α)
]
= (40)
= Γ(α)
2
n
2
−2αΓ(n
2
−α)
(S − i0)n2−α−1 , (41)
for α 6= −k, k = 0, 1, 2, . . .
Similarly, for n even and α 6= −k, k = 0, 1, 2, . . ., we have
H{fα(t− i0)} = H{(t− i0)α−1} =
=
Γ(α)
2
n
2
−2αΓ(n2 − α)
(S + i0)
n
2
−α−1 . (42)
In particular, for the case α = −k, k = 0, 1, 2, . . ., and taking into account the formula
(23) and (32) we get the following result,
H{δ(k)(t)} = 1
2
n
2
+2k
R+n
2
+k(s) =
1
2
n
2
+2kΓ(n2 + k)
S
n−2
2
+k
+ . (43)
The formula (43) was proved in [6], page 28, formula (I,5,15) by S.E. Trione.
We observe that if we consider the following family of generalized functions Nα(x± i0)
defined by the formula
Nα(x± i0) = fα(x± i0)
Γ(α)
, (44)
where fα(x ± i0) is defined by (15) and taking into account the formulae (41) and (42),
then the Hankel distributional transform of Nα(x± i0) can be expressed by the formula,
H{Nα(t± i0)} = 1
2
n
2
−2α
Nn
2
−α(S ∓ i0) (45)
for n even and α 6= −k, k = 0, 1, 2, . . .
On the other hand, the generalized function (x± i0)λ are entire of λ ([3], page 60).
In the points λ = −k, k = 1, 2, . . ., from [3], page 94, the following formula is valid,
(x± i0)−k = x−k ∓ pii(−1)
k−1
(k − 1)! δ
(k−1)(x) , (46)
therefore for the case α = −k, k = 0, 1, 2, . . . taking into account the formulae (23) and
(24) we have,
lim
α→−k
Nα(x± i0) = 0 , (47)
where the limit is in the sense of generalized functions.
the hankel transform of causal distributions 19
3 The distributional Hankel transform of
(m2 + P )α−1+
Γ(α)
,
(m2 + P )α−1−
Γ(α)
,
|m2 + P |α−1
Γ(α
2
)
,
|m2 + P |α−1sgn(m2 + P )
Γ(α+12 )
and (m2 + P ± i0)α−1.
In this paragraph we are going to extend the results (32), (34), (36), (37), (41) and (42) to
certain kinds of n-dimensional distributions called “causal” and “anticausal” distributions.
We begin with some definitions. Let x = (x1, x2, . . . , xn) be a point of the n-dimensional
euclidean space Rn. Consider a non degenerate quadratic form in n variables of the form
P = P (x) = x21 + · · · + x2µ − x2µ+1 − · · · − x2µ+ν , (48)
where µ+ ν = n (dimension of the space).
The distribution (P ± i0)λ are defined by
(P ± i0)λ = lim
ε→0
(P ± iε|x|2)λ , (49)
where
ε > 0, |x|2 = x21 + · · · + x2n (50)
and λ complex number.
The distributions (m2+P ± i0)λ are defined in an analogue manner as the distribution
(P ± i0)λ. Let us put (cf. [3], page 289)
(m2 + P ± i0)λ = lim
ε→0
(m2 + P ± i0|x|2)λ , (51)
where ε is an arbitrary positive number and m is positive real number.
It is useful to state an equivalent definition of the distribution (m2+P ± i0)λ. In this
definition appear the distributions
(m2 + P )λ+ =
{
(m2 + P )λ if (m2 + P ) ≥ 0
0 if m2 + P < 0
, (52)
and
(m2 + P )λ− =
{
0 if m2 + P > 0
(−m2 − P )λ if m2 + P ≤ 0 . (53)
From [4], page 566, we have
fλ(m
2 + P ± i0) = (m2 + P )λ+ + e±λpii(m2 + P )λ− , (54)
and from this formula we conclude immediately that
(m2 + P ± i0)λ = (m2 + P − i0)λ = (m2 + P )λ
when λ = k positive integer.
20 m. aguirre
We observe that (m2 + P ± i0)λ are entire distributional functions of λ. This is the
principal difference between the distributions, formally analogue (P ± i0)λ which have
poles at the points λ = −n2 − k, k = 0, 1, 2, . . ..
It can be proved ([4], page 573, formula (2.14) and page 575, formula (3.5)) that
(m2 + P ± i0)−k = Pf(m2 + P )−k ∓ pii(−1)
k−1
(k − 1)! δ
(k−1)(m2 + P ) , (55)
where Pf means finite part or regular part of the Laurent expansion of (m2 + P )λ+ about
λ = −k, namely
Pf(m2 + P )−k = lim
λ→−k
d
dλ
[(λ+ k)(m2 + P )λ+] . (56)
The formula (55) is a multidimensional analogue of the well-known unidimensional
formula (see formula (46)).
By causal (anticausal) distribution, we mean distributions of the form fa(n
2+P ± i0)
where (m2 + P ± i0)λ is defined by the formula (51).
On the other hand, from [7], page 6, formulae (20) and (21), we have
|m2 + P |λ = (m2 + P )λ+ + (m2 + P )λ− (57)
[sgn (m2 + P )]|m2 + P |λ = (m2 + P )λ+ − (m2 + P )λ− (58)
and from [4], page 566, we have
Resλ=−k(m
2 + P )λ+ =
(−1)k−1
(k − 1)! δ
(k−1)(m2 + P ) (59)
and
Resλ=−k(m
2 + P )λ− =
1
(k − 1)! δ
(k−1)(m2 + P ) . (60)
Therefore, from (52), (53), (57), (58) and taking into account (59) and (60) we have
lim
α→−k
R+α (m
2 + P ) = δ(k)(m2 + P ) , (61)
lim
α→−k
R−α (m
2 + P ) = (−1)kδ(k)(m2 + P ) , (62)
lim
α→−2k
Yα(m
2 + P ) =
(−1)kk !
(2k)!
δ(2k)(m2 + P ) , (63)
and
lim
α→−2k−1
Zα(m
2 + P ) =
(−1)k+1k !
(2k + 1)!
δ(2k+1)(m2 + P ) , (64)
where
R+α (m
2 + P ) =
(m2 + P )α−1+
Γ(α)
, (65)
R−α (m
2 + P ) =
(m2 + P )α−1−
Γ(α)
, (66)
the hankel transform of causal distributions 21
Yα(m
2 + P ) =
|m2 + P |α−1
Γ(α2 )
, (67)
and
Zα(m
2 + P ) =
|m2 + P |α−1sgn(m2 + P )
Γ(α+12 )
. (68)
In order to give a sense to formulae (61), (62), (63) and (64) (or “to regularize” them)
we deal with multi-dimensional generalization obtained by means of a change of variable
(see formula (7)).
Taking into account the formulae (8) and (9) we are going to obtain the Hankel dis-
tributional transform of the family of generalized functions defined by the formulae (65),
(66), (67), (68) and (54).
In fact, from (8) and (9) we have
〈H{Φ(u(x)}, ϕ〉 = 〈H{v(t)}, ψ(s)〉 (69)
where ψ(s) is defined by the formula (10).
From (32) and (69) we have,
〈H{R+α (m2 + P )}, ϕ〉 = 〈H{R+α (t)}, ψ(s)〉 =
=
1
2
n
2
−2α
〈R+n
2
−α
(s), ψ(s)〉 =
=
1
2
n
2
−2α
1
Γ(n2 − α)
〈S
n
2
−α−1
+ , ψ(s)〉 , (70)
where Sλ+ is defined in (16).
Taking into account (10) and (7), we have
〈S
n
2
−α−1
+ , ψ(s)〉 = 〈(m2 +Q)
n
2
−α−1
+ , ϕ〉 , (71)
where Q = Q(y) = g21 + · · ·+ y2µ − y2µ+1 − · · · − y2µ+ν ,\ µ+ ν = d (dimension of the
space), and\ (m2 +Q)λ+ is defined in (52).
From (70) and (71) we have proved our first basic formula:
H{R+α (m2 + P )} =
1
2
n
2
−2α
R+n
2
−α
(m2 +Q) . (72)
Similary, from (34) and (69) we have
〈H{R−α (m2 + P )}, ϕ〉 = 〈H{R−α (t)}, ψ(s)〉 =
=
(−1)n2
2
n
2
−2αΓ(n2 − α)
〈S−α+
n−2
2
− , ψ(s)〉 , (73)
where Sλ− is defined in (17).
Taking into account (10) and (7), we have
〈S−α+
n−2
2
− , ψ(s)〉 = 〈(m2 +Q)
n
2
−α−1
− , ϕ〉 , (74)
22 m. aguirre
where (m2 +Q)λ is defined in (53).
From (73) and (74) we have proved our second basic formula:
H{R−α (m2 + P )} =
(−1)n2
2
n
2
−2α
R−n
2
−α
(m2 +Q) . (75)
On the other hand, from (67), (36), (69) and taking into account the formulae (71)
and (74) we have
〈H{Yα(m2 + P )}, ϕ〉 = 〈H{Yα(t), ψ(s)}〉 =
= a(α, n, pi)[〈R+n
2
−α
(s) + (−1)n2R−n
2
−α
(s), ψ(s)〉] =
= a(α, n, pi)[〈R+n
2
−α
(s), ψ(s)〉 + (−1)n2 〈R−n
2
−α
(s), ψ(s)〈] =
= a(α, n, pi)[〈R+n
2
−α
(m2 +Q), ϕ〉 + (−1)n2 〈R−n
2
−α
(m2 +Q), ϕ〉] =
= a(α, n, pi)[〈R+n
2
−α
(m2 +Q) + (−1)n2R−n
2
−α
(m2 +Q), ϕ〉] ; (76)
where
a(α, n, pi) =
Γ(α+12 )
2
n
2
−3α+1pi
1
2
. (77)
Therefore, from (76) we obtain the following formula:
H
{ |m2 + P |α−1
Γ(α2 )
}
= a(α, n, pi)[R+n
2
−α
(m2 +Q) + (−1)n2R−n
2
−α
(m2 +Q)] , (78)
where a(α, n, pi) is defined by the formula (77).
Similarly, from (68), (37) and taking into account the formulae (71) and (74), we have
〈H{Zα(m2 + P )}, ϕ〉 = 〈H{Zα(t)}, ψ(s)〉 =
= b(α, n, pi)[〈R+n
2
−α
(s)− (−1)n2R−n
2
−α
(s), ψ(s)〉] =
= b(α, n, pi)[〈R+n
2
−α
(s), ψ(s)〉 − (−1)n2 〈R−n
2
−α
(s), ψ(s)〉] =
= b(α, n, pi)[〈R+n
2
−α
(m2 +Q), ϕ〉 − (−1)n2 〈R−n
2
−α
(m2 +Q), ϕ〉] =
= b(α, n, pi)[〈R+n
2
−α
(m2 +Q)− (−1)n2R−n
2
−α
(m2 +Q), ϕ〉] ; (79)
where
b(α, n, pi) =
Γ(α2 )
2
n
2
−3α+1pi
1
2
. (80)
Therefore, from (79) we obtain the following formula:
H
{ |m2 + P |α−1sgn (m2 + P )
Γ(α+12 )
}
=
= b(α, n, pi)[〈R+n
2
−α
(m2 +Q)− (−1)n2R−n
2
−α
(m2 +Q)] , (81)
where b(α, n, pi) is defined by the formula (80).
the hankel transform of causal distributions 23
Finally, to obtain the Hankel transform of fα(m
2+P ± i0), we shall take into account
the results (72) and (75).
In fact, for n even, from (54) and taking into account the formulae (72) and (75) we
have
H{fα(m2 + P ± i0)} = Γ(α)[H{R+α (m2 + P )}+ e±(α−1)pii .
.H{R−α (m2 + P )}] =
Γ(α)
2
n
2
2α
R+n
2
−α
(m2 +Q) +
+
e±(α−1)piiΓ(α)
2
n
2
−2α
R−n
2
−α
(m2 +Q) =
=
Γ(α)
2
n
2
−2αΓ(n2 − α)
(m2 +Q∓ i0)n2−α−1 =
=
Γ(α)
2
n
2
−2αΓ(n2 − α)
fn
2
−α(m
2 +Q∓ i0) . (82)
Therefore, from (82) we obtain the following formula:
H{(m2 + P ± i0)α−1} = Γ(α)
2
n
2
−2αΓ(n2 − α)
(m2 +Q∓ i0)n2−α−1, (83)
α 6= −k, k = 0, 1, 2, . . . , and n even. In particular, for the case α = −k, k = 0, 1, 2, . . .,
and taking into account the formula (61), from (72) we get the following result
H{δ(k)(m2 + P )} = 1
2
n
2
+2k
R+n
2
+2k(m
2 +Q) =
=
1
2
n
2
−2kΓ(n2 + k)
(m2 +Q)
n−2
2
+k
+ . (84)
The formula (84) was proved in [5], page 3, formula (36).
We observe that if consider the following family generalized functions Nα(m
2+P ± i0)
defined by the formula
Nα(m
2 + P ± i0) = fα(m
2 + P ± i0)
Γ(α)
, (85)
where fα(m
2 + P ± i0) is defined by (54), and taking into account the formula (82), then
the Hankel distributional transform of Nα(m
2 + P ± i0) can be expressed by the formula
H{Nα(m2 + P ± i0)} = 1
2
n
2
−2α
Nn
2
−α(m
2 +Q∓ i0) , (86)
for n even and α 6= −k, k = 0, 1, 2, . . ..
For the case α = −k, taking into account the formulae ((61) and ((62), we have
lim
α→−k
Nα(m
2 + P ± i0) = 0 , (87)
where the limit is in the sense of generalized functions.
24 m. aguirre
References
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[2] Leray, J. (1952) Hyperbolic Differential Equations. Mimeographed Lecture Notes, Inst.
for Advanced Studies, Princeton.
[3] Gelfand, I.M.; Shilov, G.E. (1964) Generalized Functions. Vol I, Academic Press, New
York.
[4] Bresters, D.W. (1968) “On distributions connected with quadratic forms”, SIAM J.
Appl. Math. 16: 563–581.
[5] Aguirre, M.A.; Trione, S.E. “The distributional Hankel transform of δ(k)(m2 + P )”,
Studies in Applied Mathematics 83: 111–121, Massachusetts Institute of Technology
Publ., Elsevier Science Publ. Co. Inc.
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