IJMMS 25:4 (2001) 253–263
PII. S0161171201003507
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
NEW INVERSION FORMULAS FOR THE KRÄTZEL
TRANSFORMATION
DOMINGO ISRAEL CRUZ-BÁEZ and JOSÉ RODRÍGUEZ EXPÓSITO
(Received 9 June 1999)
Abstract. We study in distributional sense by means of the kernel method an integral
transform introduced by Krätzel. It is well known that the cited transform generalizes to
the Laplace and Meijer transformation. Properties of analyticity, boundedness, and inversion theorems are established for the generalized transformation.
2000 Mathematics Subject Classification. Primary 44A15, 46F12.
1. Introduction. In this paper, we study the following integral transform,
∞
ρ Zρν (xy)f (y) dy, x > 0,
Kν f (x) =
0
(1.1)
where Zρν (x) denotes the function
Zρν (x) =
∞
0
t ν−1 e−t
ρ −x/t
dt
(1.2)
ρ
with ρ > 0, ρ ∈ N, ν ∈ C. The Kν transform is reduced to the Meijer transform when
ρ = 1 and to the Laplace transform when ρ = 1 and ν = ±1/2. Zemanian realized a
wide study of the Laplace and Meijer transformations, in distribution spaces (cf. [12,
ρ
13]). Later, Krätzel, introduced the Kν transformation, which generalizes the Meijer
transform, and in a series of papers, investigated it in the classical sense (see [5,
ρ
6]). In [8], Rao and Debnath investigated the Kν transformation on certain spaces of
distributions by means of the kernel method. Recently, the cited transformation is
studied in [1] by the adjoint method on the McBride’s spaces.
In [3], the study of Krätzel is continued obtaining Abelian, Tauberian theorems, and
some inversion formulas in the classical sense. Moreover, we can emphasize in [2] the
ρ
work developed by the authors studying the Kν transform on weighted Lp spaces,
improving a result of [4].
ρ
Motivated by the cited papers, we accomplish a study of the Kν transform by means
of the kernel method on the space of distributions of compact support.
By E(I) we denote the infinitely differentiable functions φ(t), t ∈ I = (0, ∞), such
that for all compact K we have
dk
γk (φ) = sup k φ(t) < ∞,
(1.3)
dt
t∈K
for every k ∈ N. By E (I) we denote the dual space of E(I). Moreover, by D(I), D (I)
denote the spaces of functions and distributions that can be found in [9, 11].
254
D. I. CRUZ-BÁEZ AND J. RODRÍGUEZ
Consider the following useful properties.
In [1] we see that
dk ν
Z (x) = (−1)k Zρν−k (x).
(1.4)
dx k ρ
By (1.4) and the asymptotic behaviour of Zρν (x), we deduce that for certain positive
constant α1 and k ∈ N, we have
dk ρ/(ρ+1)
ν
,
(1.5)
Z (x) ≤ α1 · x (2ν−ρ)/(2ρ+2) e−x
dx k ρ
for all x ∈ K, K compact and ν ∈ C.
Moreover, by [5] we have that if ρ ∈ N,
Aρ,ν Zρν (x) = (−1)ρ Zρν (x),
where
Aρ,ν = −ρ −1 · x ν−ρ+1
(1.6)
d ρ−ν dρ
x
.
dx
dx ρ
(1.7)
k
On the other hand, Bρ,ν,k denotes Bρ,ν,k = C1 · y −ρk−1 Bρ,ν
, where
Bρ,ν = x −ν+ρ+1
C1 =
d ρ+ν dρ
x
,
dx
dx ρ
(ρk)ρk−(2ν−ρ)/(2ρ+2)−1/(ρ+1)+1 ρ
.
2ν−ρ
ρk−(2ν−ρ)/(2ρ+2)−1/(ρ+1)+1+ν
1
Γ ρk − 2ρ+2 − ρ+1
+1 Γ
ρ
(1.8)
∞
The Mellin transform is defined by M f (s) = 0 x s−1 f (x)dx and the Mellin transform of the kernel is given by
ν
s +ν
1
,
(1.9)
M Zρ (x) (s) = Γ (s)Γ
ρ
ρ
if Re s + min(0, Re ν) > 0.
ρ
2. The generalized Kν -transform on E (I). Let ν ∈ C and ρ ∈ N. For every f ∈
ρ
E (I) we define the generalized Kν transform by the relation
ρ F (x) = Kν f (x) = f (t), Zρν (xt) .
(2.1)
It is easy to see that Zρν (xt) ∈ E(I) for x fixed, x > 0. Furthermore, if f is a locally
absolutely integrable function, then the generalized transform of f is reduced to the
ρ
classic Kν transform.
Proposition 2.1. The operators Aρ,ν and Bρ,ν define a continuous linear mapping
from E(I) into itself.
Proof. It is established without difficulty that
ρ+1
γk Aρ,ν φ ≤ C ·
γk+j (φ),
j=1
for every φ ∈ E(I).
ρ+1
γk Bρ,ν φ ≤ C ·
γk+j (φ),
j=1
(2.2)
NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION
255
We define the generalized operator A∗
ρ,ν on E (I) as the adjoint operator of Aρ,ν ,
that is,
∗
Aρ,ν f , φ = f , Aρ,ν φ ,
(2.3)
with f ∈ E (I) and φ ∈ E(I). Moreover, by Proposition 2.1, A∗
ρ,ν is a continuous linear
mapping from E (I) into itself. Note that the same occurs for the operator Bρ,ν .
ρ
Next are established some properties of the generalized Kν transformation.
ρ
Proposition 2.2. Let f ∈ E (I). If F (x) denotes the generalized Kν transform of f ,
then F (x) is infinitely differentiable on (0, ∞) and
dr
∂r ν
F
(x)
=
f
(t),
Z
(xt)
, x > 0, r ∈ N.
(2.4)
ρ
dx r
∂x r
Proof. Consider h an arbitrary increment in x > 0. Assume, without loss of generality, that 0 < |h| < (x/2)ρ/(ρ+1) .
It is easy to see that
1 ν
F (x + h) − F (h)
= f (t),
Zρ t(x + h) − Zρν (tx) .
(2.5)
h
h
We must see that
ϕh (x, t) =
1 ν
∂ ν
Z t(x + h) − Zρν (tx) −
Z (xt) → 0
h ρ
∂x ρ
(2.6)
as h → 0, in the sense of the convergence in E(I), since the result is obtained for k = 1,
by (2.5) and the continuity of f (t).
For every r ∈ N we can write
r
1 x+h u ∂ 2
∂
∂r
ν
ϕ
(x,
t)
=
Z
(ty)
dy du
h
2
∂t r
h x
∂t r ρ
x ∂y
dr
1 x+h u ∂ 2
r
ν
Z (ty) dy du
y
=
2
h x
d(ty)r ρ
x ∂y
dr +1
dr
1 x+h u
Z ν (ty) + 2r y r −1 t
Z ν (ty)
r (r − 1)y r −2
=
r ρ
r +1 ρ
h x
d(ty)
d(ty)
x
dr +2
r 2
ν
+y t
Z (ty) dy du.
d(ty)r +2 ρ
(2.7)
By virtue of (1.5), given t ∈ K, K compact, y > x/2, it follows that there exists a
positive constant C such that
dr +j ρ/(ρ+1)
ν
, for j ∈ N.
(2.8)
Z (ty) ≤ C(yt)(2ν−ρ)/(2ρ+2) · e−(tx/2)
d(ty)r +j ρ
Therefore
r
∂
≤ C · t (2ν−ρ)/(2ρ+2) · e−(tx/2)ρ/(ρ+1)
ϕ
(x,
t)
∂t r h
1 x+h u r −2
·
+ y r −1 t + y r t 2 dy du
y
h x
x
ρ/(ρ+1)
≤ C · e−α·(tx)
for all t ∈ K and 0 < |h| < (x/2)ρ/(ρ+1) , being α a suitable constant.
(2.9)
256
D. I. CRUZ-BÁEZ AND J. RODRÍGUEZ
Then given $ > 0, there exists t0 ∈ K = [a, b] such that
r
∂
<$
ϕ
(x,
t)
∂t r h
(2.10)
for t > t0 and 0 < |h| < x ρ/(ρ+1) /2.
Furthermore, for each t ∈ [a, t0 ] we have
r
1 x+h u ∂
r −2
r −1
r 2
+y
t + y t dy du .
y
∂t r ϕh (x, t) ≤ C · h x
x
(2.11)
Then (∂ r /∂t r )ϕh (x, t) → 0 as h → 0 uniformly in t ∈ [a, t0 ]. Therefore it is concluded that γr (ϕh (x, t)) → 0 as h → 0.
By proceeding inductively the result follows.
ρ
Proposition 2.3. Let f ∈ E (I) and F (x) = (Kν f )(x) for x > 0, then
F (x) ≤ C · x r +(2ν−ρ)/(2ρ+2) · e−α x ρ/(ρ+1) ,
x > 0,
(2.12)
being C and α suitable constants and r ∈ N.
Proof. We know by [13, Theorem 1.8-1, pages 18–19] that there exists r ∈ N such
that
F (x) ≤ C max γk Z ν (xt) ∀x > 0.
(2.13)
ρ
0≤k≤r
By the asymptotic behaviour we have for every K compact,
dk ν
F (x) ≤ C max sup x k
(xt)
Z
d(xt)k ρ
0≤k≤r t∈K
ρ/(ρ+1) ≤ C max sup x k (xt)(2ν−ρ)/(2ρ+2) e−(xt)
0≤k≤r t∈K
ρ/(ρ+1) ≤ C · x r +(2ν−ρ)/(2ρ+2) · max sup t (2ν−ρ)/(2ρ+2) e−(xt)
(2.14)
0≤k≤r t∈K
≤ C ·x
r +(2ν−ρ)/(2ρ+2) −αx ρ/(ρ+1)
e
.
With the following proposition we obtain an operational formula for the generalized
ρ
Kν transform, that includes the operator A∗
ρ,ν , ρ ∈ N.
Proposition 2.4. Let P be a polynomial, if f ∈ E (I), then
ρ ρ ∗ Kν P Aρ,ν f (x) = P (−x)ρ Kν f (x)
(2.15)
for x > 0 and ρ ∈ N.
Proof. By (1.6) and according to Proposition 2.1 it follows that
∗ ρ ν
Kν P A∗
ρ,ν f (x) = P Aρ,ν f (t), Zρ (xt)
= f (t), P Aρ,ν Zρν (xt)
= f (t), P (−x)ρ Zρν (xt)
ρ = P (−x)ρ Kν f (x).
(2.16)
NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION
257
ρ
Now we establish an inversion theorem for the Kν - transform using a similar procedure to employ by Malgonde and Saxena [7].
Lemma 2.5. Let f ∈ E (I). Then
∞
∞
x −s f (t), Zρν (xt) dx = f (t),
x −s Zρν (xt)dx .
0
0
(2.17)
Proof. Let N ∈ N. First, we prove that
N
0
N
x −s f (t), Zρν (xt) dx = f (t),
x −s Zρν (xt)dx .
0
(2.18)
We take {xr ,l }lr =0 a partition of the interval [0, N] such that dl = xr ,l − xr −1,l for
each r = 1, 2, . . . , l. Then we can write
N
0
dl ·
l
x −s f (t), Zρν (xt) dx = lim dl ·
xr−s,l f (t), Zρν xr ,l t ,
l
r =0
l →∞
xr−s,l
f (t), Zρν
xr ,l t
r =0
= f (t), dl
l
r =0
xr−s,l Zρν
xr ,l t
(2.19)
.
Then (2.18) is established if we demonstrated
lim dl
l→∞
l
r =0
xr−s,l · Zρν xr ,l t =
N
0
x −s Zρν (xt) dx
in the sense of the convergence in E(I).
Then, since the function

k
ν

x −s+k d
Zρ (xt) if x ∈ [0, N],
k
d(xt)
g(t, x) =

0
if x = 0,
(2.20)
(2.21)
with t ∈ [a, b], 0 < a < b < ∞, is uniformly continuous on (t, x) ∈ [a, b] × [0, N], we
have that if $ > 0 and m ∈ N, there exists l0 ∈ N such that


k
N
l
d
−s ν
−s ν

sup k dl
xr ,l Zρ xr ,l t −
x Zρ (xt) dx 
(2.22)
<$
dt
0
t∈K r =0
for l > l0 . Therefore we obtain (2.20).
Using (1.5), for m, k ∈ N, we have that
∞
dk ∞
ρ/(ρ+1)
k
x −s Zρν (xt)dx ≤ C · t (2ν−ρ)/(2ρ+2)
x − Re s+(2ν−ρ)/(2ρ+2) e−(xt)
dx → 0
dt N
N
(2.23)
uniformly in t ∈ K, as N → ∞.
Moreover, by Proposition 2.3 we have
∞
∞
ρ/(ρ+1)
x −s f (t), Zρν (xt) dx x −s+r +(2ν−ρ)/(2ρ+2) e−αx
dx,
(2.24)
≤C
N
being α a suitable constant.
N
258
D. I. CRUZ-BÁEZ AND J. RODRÍGUEZ
Then
lim
∞
N→∞ N
x −s f (t), Zρν (xt) dx = 0.
(2.25)
Hence by (2.18), (2.23), and (2.25) we obtain (2.17).
The following lemma is obtained from [7, Lemmas 2 and 3] with a slight modification.
Lemma 2.6. Let φ ∈ D(I), we denote
ψ(s) =
∞
0
y −s φ(y) dy.
(2.26)
Then
R
R
(i) −R f (u), uσ+iw−1 ψ(σ+iw)dw = f (u), −R uσ+iw−1 ψ(σ+iw)dw with σ > 0
and R > 0,
∞
(ii) (1/π ) 0 φ(y)(u/y)σ (sin(R log(u/y))/u log(u/y)) dy → φ(u) as R → ∞, in
the sense of the convergence in E(I), with σ > 0.
Now we demonstrate the first inversion theorem.
Theorem 2.7. Let f ∈ E (I), φ ∈ D(I), ν ∈ C and σ < 1 + min(0, Re ν). Then
lim
R→∞
1
2π i
σ +iR
σ −iR
1
y −s
K(s)
∞
0
x
−s
F (x) dx ds, φ(y)
= f (t), φ(t) ,
(2.27)
ρ
where F (x) = (Kν f )(x), for x > 0, and
K(s) =
1
Γ (1 − s)Γ
ρ
1−s +ν
ρ
.
(2.28)
Proof. Given f ∈ E (I), we denote F (x) = f (t), Zρν (xt), for x > 0. It is easy to
see that the function
ϕR (y) =
1
2π i
σ +iR
σ −iR
1
y −s
K(s)
∞
0
x −s F (x) dx ds
(2.29)
is continuous for y > 0, R > 0. Therefore, ϕR (y) defines a regular distribution in
D (I) being
ϕR (y), φ(y) =
1
2π i
∞
0
φ(y)
σ +iR
σ −iR
1
y −s
K(s)
∞
0
x −s F (x) dx ds dy
(2.30)
for all φ ∈ D(I).
By the Fubini’s theorem we can interchange the order of integration and we can
write
ϕR (y), φ(y) =
1
2π i
σ +iR
σ −iR
1
K(s)
∞
0
x −s f (t), Zρν (xt) dx
∞
0
y −s φ(y) dy ds
(2.31)
NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION
By Lemma 2.5, we get
σ +iR
∞
∞
1
f (t), t s−1
u−s Zρν (u) du
y −s φ(y)dyds
σ −iR K(s)
0
0
σ +iR
∞ −s
1
y φ(y) dy ds.
f (t), t s−1
=
2π i σ −iR
0
(2.32)
ϕR (y), φ(y) =
259
1
2π i
Then Lemma 2.6(i) permits us to obtain
σ +iR
∞
1
s−1
t
y −s φ(y) dy ds .
ϕR (y), φ(y) = f (t),
2π i σ −iR
0
(2.33)
Finally, interchanging the order of integration and using Lemma 2.6(ii) we achieve
sin R log(u/y)
1 ∞ u σ
dy → f (t), φ(t)
ϕR (y), φ(y) = f (t),
φ(y)
π 0 y
u log u/y
(2.34)
as R → ∞. With this, Theorem 2.7 is demonstrated.
By Theorem 2.7 the following uniqueness theorem is deduced.
ρ
ρ
Theorem 2.8. Let f , g ∈ E (I) and ν ∈ C. If (Kν f )(x) = (Kν g)(x), for x > 0, then
f = g in the sense of an equality in D (I).
Proof. It is sufficient to see that for each φ ∈ D(I), f (t) − g(t), φ(t) = 0.
This fact is established immediately since,
f (t) − g(t), φ(t)
σ +iR
∞
ρ ρ 1
1
y −s
x −s Kν f (x) − Kν g (x) dx ds, φ(y) = 0,
= lim
R→∞ 2π i σ −iR K(s)
0
(2.35)
where σ and K(s) are as in Theorem 2.7.
In the following theorem we obtain another inversion formula, in which appears a
generalization of an operator of Post-Widder type, being obtained as a particular case
an inversion formula for the Laplace and Meijer transformation.
ρ
Theorem 2.9. Let f ∈ E (I) and Re ν ≥ ρ/2 − 1. If F (x) denotes the Kν generalized
transform of f , then
ρk
∗
lim Bρ,ν,k F
, φ(y) = f (t), φ(t)
(2.36)
y
k→∞
∗
is the adjoint operator of Bρ,ν,k .
for each φ ∈ D(I), where Bρ,ν,k
Proof. Let f ∈ E (I) and F (x) = f (t), Zρν (xt), for x > 0.
By Proposition 2.1 and (1.6) we have
ρk
ρkt
∗
−ρk−1
k
ν
Bρ,ν,k F
= C1 y
f (t), Bν,ρ Zρ
y
y
ρkt
−ρk−1
k
ρk ν
= C1 y
,
f (t), ρ (ρkt) Zρ
y
C1 being introduced in Section 1.
(2.37)
260
D. I. CRUZ-BÁEZ AND J. RODRÍGUEZ
∗
Moreover, Bρ,ν,k
F (ρk/y) defines a regular distribution in D (I) by
∗
F
Bρ,ν,k
∞
ρk
ρk
∗
, φ(y) =
φ(y) dy,
Bρ,ν,k F
y
y
0
∀φ ∈ D(I).
(2.38)
We must see that
∗
Bρ,ν,k
F
∞
ρk
ρkt
, φ(y) = f (t), C1 t ρk
φ(y) dy
y −ρk−1 Zρν
y
y
0
(2.39)
for every φ ∈ D(I).
Consider 0 < a < b < ∞ such that φ(x) = 0, x ∈ [a, b], φ ∈ D(I).
We denote by {ym,l }lm=0 a partition of [a, b] with dl = ym,l − ym,l−1 , m = 1, 2, . . . , l,
then we achieve
∞
0
∗
Bρ,ν,k
F
ρk
φ(y) dy
y
= lim dl
l→∞
m=0
= lim
l→∞
l
∗
Bρ,ν,k
F
f (t), t
ρk
ρk
ym,l
k
φ(xm,l )
l
ρk
C1 ρ (ρk) dl
(2.40)
m=0
−ρk−1
ym,l Zρν
ρkt
ym,l
φ xm,l
.
Therefore we must see that
lim dl
l→∞
l
m=0
−ρk−1
ym,l
Zρν
ρkt
ym,l
φ ym,l =
b
a
y −ρk−1 Zρν
ρkt
y
φ(y) dy
(2.41)
in the sense of the convergence in E(I).
Consider K ⊂ (0, ∞) compact and r ∈ N.
Then we have


b
l
dr  −ρk−1 ν ρkt
ρkt
−ρk−1 ν
φ(y) dy 
ym,l Zρ
y
Zρ
dl
φ ym,l −
dt r
ym,l
y
a
m=0
b
l
r
r
ρkt
ρkt
−ρk−1 d
ν
−ρk−1 d
ν
φ(y) dy.
ym,l
y
−
= dl
Z
φ
y
Z
m,l
ρ
ρ
dt r
ym,l
dt r
y
a
m=0
(2.42)
Hence, since the function y −ρk−1 (dr /dt r )(Zρν (ρkt/y))φ(y) is uniformly continuous
for (x, y) ∈ [a, b] × K, then
lim dl
l→∞
l
m=0
−ρk−1
ym,l Zρν
ρkt
ym,l
φ(xm,l ) =
b
a
y
−ρk−1
uniformly in y ∈ K. Thus (2.39) is demonstrated.
Zρν
ρkt
y
φ(y) dy
(2.43)
NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION
On the other hand, making a change of variables we obtain
∞
∞
ρk
ρku
∗
φ(y) dy = g(t), C1 t −ρk−1
ψ(u) du
Bρ,ν,k
F
uρk Zρν
y
t
0
0
with g(t) = (1/t)f (1/t) and ψ(u) = (1/u)φ(1/u).
To complete the demonstration we see that
∞
ρku
lim C1 t −ρk−1
ψ(u) du = ψ(t)
uρk Zρν
t
k→∞
0
in the sense of the convergence in E(I).
Now, according to (1.9) we have
∞
ρku
du
uρk−((2ν−ρ)/(2ρ+2))−(1/(ρ+1)) Zρν
t
0
(ρ+1)k−((2ν−ρ)/(2ρ+2))−(1/(ρ+1))+1
1
2ν − ρ
t
−
+1
ρ −1 Γ ρk −
=
ρk
2ρ + 2 ρ + 1
ρk − ((2ν − ρ)/(2ρ + 2)) − (1/(ρ + 1)) + 1 + ν
·Γ
.
ρ
261
(2.44)
(2.45)
(2.46)
Then, we can deduce that
∞
d
−ρk−1
ρk ν ρku
ψ(u)
du
−
ψ(t)
t
u
Z
C
1
ρ
dt r
t
0
∞
−ρk−1
ρk−((2ν−ρ)/(2ρ+2))−(1/(ρ+1)) ν ρku
= C1 t
u
Zρ
t
0
r
r
d
u
((2ν−ρ)/(2ρ+2))+(1/(ρ+1))
((2ν−ρ)/(2ρ+2))+(1/(ρ+1)) d
· u
ψ(u)−t
ψ(t) du
t du
dt r
(2.47)
r
for every r ∈ N.
Therefore by (1.5) we have that for all t ∈ K, K ⊂ (0, ∞), K compact, there exists a
constant C > 0 such that
∞
dr
ρkt
r C · t −(ρ+1)k−1
z(ρ+1)k ψ(z)dz − ψ(t) Zρν
dt
y
0
ρk−k+1−1/(ρ+1)
∞
ρ
(ρk)
(2.48)
t −ρk−ρ/(ρ+1)
z(ρ+1)k e−α2 (ρkz/t)
≤C
((ρ + 1)k)!
ρ +1 0
r
z d
dr
· z(2ν−ρ)/(2ρ+2)
ψ(z) − t (2ν−ρ)/(2ρ+2) r ψ(t) dz.
t dz
dt
We divided the last integral in this way

∞
((ρ+1)k)1/ρ (1−η)(ρ+1)/ρ t(ρk)−1 ((ρ+1)k)1/ρ (1+η)(ρ+1)/ρ t(ρk)−1
C1
= C1 
+
0
0
+
∞
((ρ+1)k)1/ρ (1+η)(ρ+1)/ρ t(ρk)−1
((ρ+1)k)1/ρ (1−η)(ρ+1)/ρ t(ρk)−1
= I1 (t, k) + I2 (t, k) + I3 (t, k)
with η > 0 and C1 = C((ρk)ρk−k+1−1/(ρ+1) /((ρ + 1)k)!)t −ρk−ρ/(ρ+1) ρ/(ρ + 1).
(2.49)
262
D. I. CRUZ-BÁEZ AND J. RODRÍGUEZ
For each t ∈ K, we obtain
I1 (t, k)
(ρk)ρk−k+1−1/(ρ+1) −ρk−ρ/(ρ+1) ρ
t
ρ +1
(ρ + 1)k !
r
((ρ+1)k)1/ρ (1−η)(ρ+1)/ρ t(ρk)−1
ρk+(2ν−ρ)/(2ρ+2) −(ρku/t)ρ/(ρ+1) u d
·
u
e
ψ(u) du
t du
0
≤C
+
((ρ+1)k)1/ρ (1−η)(ρ+1)/ρ t(ρk)−1
0
ρ/(ρ+1)
uρk−1/(ρ+1) e−(ρku/t)
r
(2ν−ρ)/(2ρ+2)+1/(ρ+1) d
·t
dt r ψ(t) du
(2.50)
and making an appropriate change of variables, we obtain
(ρ+1)k+1 1−η
I1 (t, k) ≤ C (ρ + 1)k
x (ρ+1)k e−(ρ+1)kx dx,
(ρ + 1)k !
0
(2.51)
C being a suitable constant and Re ν ≥ ρ/2 − 1.
Now, using [10, Theorem (5b), page 288] we achieve
I1 (t, k) → 0,
(2.52)
as k → ∞ uniformly in t ∈ K.
Proceeding in a similar way we obtain that
I3 (t, k) → 0,
(2.53)
as k → ∞ uniformly in t ∈ K.
Then, it remains I2 (t, k), for this, using the mean value theorem we can write
r
dr
(2ν−ρ)/(2ρ+2)+1/(ρ+1) u d
u
ψ(u) − t (2ν−ρ)/(2ρ+2)+1/(ρ+1) r ψ(t) ≤ C2 |t − u|
t du
dt
(2.54)
for u, t ∈ (0, ∞) and C2 a suitable constant.
Then if u ∈ (((ρ + 1)k)1/ρ (1 − η)(ρ+1)/ρ t(ρk)−1 , ((ρ + 1)k)1/ρ (1 + η)(ρ+1)/ρ t(ρk)−1 )
we have
(ρ+1)k+1 1+η
I2 (t, k) ≤ C3 (ρ + 1)k
x (ρ+1)k e−(ρ+1)kx dx,
(2.55)
(ρ + 1)k !
1−η
applying [10, Theorem (5b), page 287] we obtain the desired result.
Therefore we can deduce that
lim
k→∞
∗
F
Cρ,ν,k
ρk
, φ(y) = g(t), ψ(t) = f (t), φ(t)
∀φ ∈ D(I).
y
(2.56)
NEW INVERSION FORMULAS FOR THE KRÄTZEL TRANSFORMATION
263
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José Rodríguez Expósito: Departamento de Análisis Matemático, Universidad de La
Laguna, 38271 La Laguna Tenerife, Spain
E-mail address: jorodriguez@ull.es
Domingo Israel Cruz Báez: Departamento de Economía Aplicada, Universidad de La
Laguna, 38071 La Laguna Tenerife, Spain
E-mail address: dicruz@ull.es