Memoirs on Differential Equations and Mathematical Physics
Volume 60, 2013, 57–72
L. P. Castro and M. M. Rodrigues
THE WEIERSTRASS–WHITTAKER
INTEGRAL TRANSFORM
Dedicated to the memory of
Professor Viktor Kupradze (1903–1985)
on the 110th anniversary of his birthday
Abstract. We introduce a Weierstrass type transform associated with
the Whittaker integral transform, which we refer to as Weierstrass–Whittaker integral transform. We examine some properties of the transform and
show, in particular, that it is helpful in solving of a generalized non-stationary heat equation with an initial condition.
2010 Mathematics Subject Classification. 44A15, 33C15, 33C20,
33C90, 35A22, 35C15, 35K15.
Key words and phrases. Weierstrass–Whittaker integral transform,
Weierstrass transform, Whittaker integral transform, heat kernel, non-stationary heat type equation.
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The Weierstrass–Whittaker Integral Transform
59
1. Introduction
The Whittaker functions Mµ,ν and Wµ,ν of first and second order have
acquired an increasing significance due to their frequent use in applications
of mathematics to physical and technical problems (cf., e.g., [2]). Moreover,
they are closely related to the confluent hypergeometric functions which play
an important role in various branches of applied mathematics and theoretical physics. For instance, this is the case in fluid mechanics, electromagnetic
diffraction theory and atomic structure theory. This justifies a continuous
effort in studying properties of these functions and in gathering information
about them, as well as the integral equations and transforms generated by
them.
For a somehow much more detailed account of several significant results on the Whittaker and Weierstrass type transforms, over the last halfcentury, we refer to [1, 3–7, 11–14].
Let us consider the integral transform
+∞
Z
xτ
1
[W f ](τ ) =
e− 2 Wµ,ν (xτ )f (x)e−(x+ x ) xα dx, τ > 0,
(1.1)
0
where α > 0. The main purpose of this work is to define an integral transform associated with the Whittaker integral transform (1.1) – which will be
called Weierstrass–Whittaker transform – and to study some of its properties and possible applications. We define such integral transform by
+∞
Z
1
[Wt f ](x) =
Kt (x, y)f (y)e−(y+ y ) y α dy,
(1.2)
0
where Kt (x, y) is the heat kernel associated with the Whittaker transform
(to be also studied later) and which is defined as
+∞
Z
yτ
2
xτ
1
Kt (x, y) =
e−4ν τ t e− 2 Wµ,ν (yτ )e− 2 Wµ,ν (xτ )e−(τ + τ ) τ α dτ
0
for t, x, y > 0.
The integral transform Wt f is a variant of the usual Weierstrass transform [9] and solves the heat type problem

∂t [Wt f ](x) = −Lx [Wt f ](x),
t, x > 0,
 lim [Wt f ](x) = f (x),
t→0
where
Lx = 4τ 3 x2
d2
d
+ 4τ 4 x2
+ τ 3 x2 (τ 2 − 1) + 4µτ 2 x + τ.
dx2
dx
60
L. P. Castro and M. M. Rodrigues
2. The Whittaker Integral Transform
In this section, we study some of the mapping properties of the integral
transform (1.1) which may, in fact, be viewed as an operator acting from
1
1
L2 (R+ , e−(x+ x ) xα dx) into L2 (R+ , e−(τ + τ ) τ α dτ ).
1
So, we consider the weighted Hilbert spaces L2 (R+ , e−(x+ x ) xα dx) endowed with the inner product
hf, gi
1
L2 (R+ ,e−(x+ x ) xα dx)
+∞
Z
1
=
f (x)g(x)e−(x+ x ) xα dx
(2.1)
0
which generates the associated norm
kf k
1
L2 (R+ ,e−(x+ x ) xα dx)
+∞
µZ
¶1/2
1
) α
2 −(x+ x
=
|f (x)| e
x dx
.
(2.2)
0
In order to prove the convergence of the integral transform (1.1), we have
the following auxiliary result.
1
Theorem 2.1. Let f ∈ L2 (R+ , e−(x+ x ) xα dx) and
©
ª
α > max 2|ν| − 2, 0 .
The integral transform (1.1) is absolutely convergent and the following uniform estimate
¯
¯
¯[W f ](τ )¯ ≤ Cµ,ν (τ )kf k
1
.
(2.3)
2
+ −(x+ x ) α
L (R ,e
x dx)
holds.
Proof. Invoking the Cauchy–Schwarz inequality and relation (2.19.24.7) in
[8], we have
¯
¯
¯[W f ](τ )¯ ≤
+∞
Z
¯
¯ − xτ
¯e 2 Wµ,ν (xτ )f (x)e−(x+ x1 ) xα ¯ dx ≤
0
+∞
µZ
¶1/2
xτ
xτ
1
≤
e− 2 Wµ,ν (xτ )e− 2 Wµ,ν (xτ )e−(x+ x ) xα dx
×
0
+∞
µZ
¶1/2
1
2 −(x+ x
) α
×
|f (x)| e
x dx
≤
0
+∞
µZ
¶1/2
− xτ
− xτ
α
2
2
≤
e
Wµ,ν (xτ )e
Wµ,ν (xτ )x dx
kf k
1
L2 (R+ ,e−(x+ x ) xα dx)
=
0
= Cµ,ν (τ ) kf k
1
L2 (R+ ,e−(x+ x ) xα dx)
,
(2.4)
61
The Weierstrass–Whittaker Integral Transform
where
µ
Γ(−2ν)Γ(α + 2ν + 2)Γ(2 + α)
×
Γ( 12 − µ − ν)Γ( 52 − µ + α + ν)
³1
´
5
+ µ + ν, 2 + α + 2ν, 2 + α; 1 + 2ν, + α + ν − µ; 1 +
× 3F 2
2
2
Γ(2ν)Γ(α − 2ν + 2)Γ(2 + α)
+ 1
×
Γ( 2 − µ + ν)Γ( 52 − µ + α + ν)
³1
´¶1/2
5
× 3F 2
,
− µ + ν, 2 + α, 2 + α − 2ν; 1 − 2ν, + α − ν − µ; −1
2
2
Cµ,ν (τ ) = τ −
α+1
2
with τ > 0, and where 3F 2 denotes the generalized hypergeometric function.
Hence, besides the estimation in question, the convergence of the integral
transform (1.1) is also obtained.
¤
We now concentrate on the image of the integral transform for the elements considered above. Namely, for that elements, in the next result we
1
obtain that W f ∈ L2 (R+ , e−(τ + τ ) τ α dτ ).
Theorem 2.2. Let α > max{2|ν| − 2, 0}.
1
If f ∈ L2 (R+ , e−(x+ x ) xα dx), then the Whittaker integral transform
1
[W f ](τ ) belongs to the space L2 (R+ , e−(τ + τ ) τ α dτ ).
1
Proof. From the definition of the norm in L2 (R+ , e−(τ + τ ) τ α dτ ), taking into
1
account that f ∈ L2 (R+ , e−(x+ x ) xα dx) and using (2.4), we obtain
kW f k2 2
1
L (R+ ,e−(τ + τ ) τ α dτ )
=
+∞
Z
¯
¯
¯[W f ](τ )¯2 e−(τ + τ1 ) τ α dτ ≤
0
+∞
Z
≤
(Cµ,ν (τ ))2 kf k2 2
1
1
L (R+ ,e−(x+ x ) xα )
e−(τ + τ ) τ α dτ =
0
=
∗
Cµ,ν
kf k2 2 + −(x+ 1 ) α
x x dx)
L (R ,e
³
+∞
Z
1
τ −(α+1) e−(τ + τ ) τ α dτ ≤
0
1´ ∗
≤ Γ(0, 1) +
C kf k2 2 + −(x+ 1 ) α ,
x x dx)
L (R ,e
e µ,ν
where
Γ(−2ν)Γ(α + 2ν + 2)Γ(2 + α)
×
Γ( 12 − µ − ν)Γ( 52 − µ + α + ν)
³1
´
5
× 3F 2
+ µ + ν, 2 + α + 2ν, 2 + α; 1 + 2ν, + α + ν − µ; 1 +
2
2
Γ(2ν)Γ(α − 2ν + 2)Γ(2 + α)
+ 1
×
Γ( 2 − µ + ν)Γ( 52 − µ + α + ν)
∗
Cµ,ν
=
(2.5)
62
L. P. Castro and M. M. Rodrigues
³1
× 3F 2
2
− µ + ν, 2 + α, 2 + α − 2ν; 1 − 2ν,
´
5
+ α − ν − µ; −1 ,
2
(2.6)
and
+∞
Z
1
τ −(α+1) e−(τ + τ ) τ α dτ =
0
Z1
=
τ
+∞
Z
1
τ dτ +
τ −(α+1) e−(τ + τ ) τ α dτ ≤
−(α+1) −(τ + τ1 ) α
e
0
1
Z1
≤
+∞
Z
1
−1 − τ1 −τ
τ e e dτ +
τ −α e−(τ + τ ) τ α dτ ≤
0
1
Z1
1
+∞
Z
1
dτ +
e−(τ + τ ) dτ ≤
1
+∞
Z
1
e−τ dτ = Γ(0, 1) + ,
dτ +
e
τ −1 e− τ
≤
0
1
Z1
τ −1 e− τ
≤
(2.7)
1
0
with Γ(a, x) denoting the incomplete Gamma function.
¤
3. The Heat Kernel Related to the Whittaker Integral
Transform
In order to introduce in a formal way the Weierstrass–Whittaker transform (1.2), we need first to study the heat kernel associated with the Whittaker transform. Therefore, we will introduce in this section the heat kernel
associated with the Whittaker integral transform. Moreover, we will define
and examine some of its properties.
Let us introduce the Hilbert space HK (R+ ), defined as the subspace of
1
2
L (R+ , e−(x+ x ) xα dx) formed by all functions f such that
1
W f ∈ L2 (R+ , e−(τ + τ ) τ α dτ ).
HK (R+ ) is endowed with the inner product
hf, giHK
+∞
Z
1
[W f ](τ )[W g](τ )e−(τ + τ ) τ α dτ
=
(3.1)
0
and, consequently, the norm of HK (R+ ) is given by
kf kHK =
p
hf, f iHK
+∞
µZ
¶1/2
¯
¯
¯[W f ](τ )¯2 e−(τ + τ1 ) τ α dτ
=
.
0
(3.2)
The Weierstrass–Whittaker Integral Transform
63
Proposition 3.1. Let α > max{2|ν| − 2, 0}. For t > 0, we introduce
Kt (x, y) defined on ]0, +∞[ × ]0, +∞[ by
Kt (x, y) =
+∞
Z
yτ
2
xτ
1
e−4ν τ t e− 2 Wµ,ν (xτ )e− 2 Wµ,ν (yτ )e−(τ + τ ) τ α dτ.
(3.3)
0
For all y ∈ ]0, +∞[ , the function
x 7→ Kt (x, y)
belongs to HK (R+ ).
Proof. Invoking the Cauchy–Schwarz inequality and the relation (2.19.24.7)
in [8], we will be able to prove first the fact that the kernel belongs to
1
L2 (R+ , e−(x+ x ) xα dx). Indeed,
kKt k2 2
1
L (R+ ,e−(x+ x ) xα dx)
=
+∞
Z
1
|Kt (x, y)|2 e−(x+ x ) xα dx =
0
+∞
+∞
¶2
µZ
Z
1
− yτ
−(τ + τ1 ) α
−4ν 2 τ t − xτ
2
2
Wµ,ν (xτ )e
Wµ,ν (yτ )e
=
τ dτ e−(x+ x ) xα dx ≤
e
e
0
0
+∞µ Z
+∞
Z
≤
(e
0
− xτ
2
¶
τ dτ ×
2 −(τ + τ1 ) α
Wµ,ν (xτ )) e
0
+∞
µZ
¶
¢2 −(τ + 1 ) α
¡ − yτ
1
τ τ
dτ e−(x+ x ) xα dx ≤
e 2 Wµ,ν (yτ ) e
×
0
+∞µ Z
+∞
¶
Z
1
xτ
≤
(e− 2 Wµ,ν (xτ ))2 τ α dτ e−(x+ x ) xα dx×
0
0
+∞
µZ
¶
¡ − yτ
¢2
×
e 2 Wµ,ν (yτ ) τ α dτ =
0
=
∗
(Cµ,ν
)2 y −(α+1)
+∞
Z
1
x−(α+1) e−(x+ x ) xα dx ≤
0
1 ´ ∗ 2 −(α+1)
(Cµ,ν ) y
,
≤ Γ(0, 1) +
e
³
(3.4)
∗
where Cµ,ν
is given by (2.6).
In order to prove that Kt ∈ HK (R+ ), we still need to prove that W Kt ∈
1
L2 (R+ , e−(τ + τ ) τ α dτ ).
64
L. P. Castro and M. M. Rodrigues
For α > max{2|ν| − 2, 0}, we obtain the following estimate by using the
Cauchy–Schwarz inequality:
+∞
¯Z
¯
¯
¯
1
− xτ
−(x+
)
α
¯
x x dx¯ ≤
e 2 Wµ,ν (xτ )Kt (x, y)e
|W Kt | = ¯
¯
0
+∞
µZ
¶1/2
xτ
1
≤
(e− 2 Wµ,ν (xτ ))2 e−(x+ x ) xα dx
×
0
+∞
µZ
¶1/2
1
×
|Kt (x, y)|2 e−(x+ x ) xα dx
≤
0
+∞
µZ
¶1/2
¡ − xτ
¢2
e 2 Wµ,ν (xτ ) xα dx
≤
kKt k
1
L2 (R+ ,e−(x+ x ) xα dx)
=
0
=
α+1
∗
(Cµ,ν
)1/2 τ − 2 kKt k 2 + −(x+ x1 ) α .
x dx)
L (R ,e
Taking into account the previous inequality, we have
kW Kt k2 2 + −(τ + 1 ) α
τ τ dτ )
L (R ,e
+∞
Z
¯
¯
¯W Kt (x, y)¯2 e−(τ + τ1 ) τ α dτ ≤
=
0
≤
∗
Cµ,ν
kKt k2 2 + −(x+ 1 ) α
x x dx)
L (R ,e
³
+∞
Z
1
τ −(α+1) e−(τ + τ ) τ α dτ ≤
0
1´ ∗
≤ Γ(0, 1) +
C kKt k2 2 + −(x+ 1 ) α . (3.5)
x x dx)
L (R ,e
e µ,ν
Therefore, we have just proved that, for y > 0, the function x 7→ Kt (x, y)
belongs to HK (R+ ).
¤
In order to obtain some important results related to the heat kernel
and the Weierstrass transform, we need to introduce a new Hilbert space
∗
which we denote by HK
(R+ ). Towards this end, we need first to guarantee
the following result (which will ensure that the above-mentioned new space
definition will be coherent with our purposes).
Lemma 3.2. If f ∈ HK (R+ ), then
+∞
Z
xτ
1
[W f ](τ )e− 2 Wµ,ν (xτ )e−(τ + τ ) τ α dτ
(3.6)
0
belongs to HK (R+ ).
Proof. Having in mind the definition of HK (R+ ), under the above hypothesis, we realize that we have to prove that both the element in (3.6) and its
1
image under W must belong to L2 (R+ , e−(x+ x ) xα dx).
65
The Weierstrass–Whittaker Integral Transform
For start, we will directly prove that for all elements f ∈ HK (R+ ) we
have
+∞
Z
¡
¢
xτ
1
1
[W f ](τ )e− 2 Wµ,ν (xτ )e−(τ + τ ) τ α dτ ∈ L2 R+ , e−(x+ x ) xα dx .
0
Indeed,
+∞
+∞¯ Z
¯2
Z
¯
¯
1
− xτ
−(τ + τ1 ) α
¯
2
[W f ](τ )e
τ dτ ¯¯ e−(x+ x ) xα dx ≤
Wµ,ν (xτ )e
¯
0
+∞µ Z
+∞
Z
0
¡
≤
0
µ
¢2
[W f ](τ ) e
0
+∞
Z
¡
×
¶
− xτ
2
e
−(τ + τ1 ) α
τ dτ ×
¶
¢2 −(τ + 1 ) α
1
τ
Wµ,ν (xτ ) e
τ dτ e−(x+ x ) xα dx ≤
0
≤
+∞µ Z
+∞
Z
¡
0
×
¶
¢2
1
[W f ](τ ) e−(τ + τ ) τ α dτ ×
0
+∞
µZ
¡
e−
xτ
2
¶
¢2
1
Wµ,ν (xτ ) τ α dτ e−(x+ x ) xα dx ≤
0
≤
∗
kW f k 2 + −(τ + τ1 ) α
Cµ,ν
L (R ,e
τ dτ )
+∞
Z
1
x−α−1 e−(x+ x ) xα dx ≤
0
∗
kW f k
≤ Cµ,ν
1
L2 (R+ ,e−(τ + τ ) τ α dτ )
+∞
Z
x−α−1 e−x xα dx ≤
0
³
1´
∗
≤ Cµ,ν
Γ(0, 1) +
kW f k 2 + −(τ + τ1 ) α .
(3.7)
L (R ,e
τ dτ )
e
From the previous inequality, taking into account the definition of the
Whittaker integral transform (1.1), we have the following inequality related
with the Whittaker transform:
+∞
¯ ·Z
¸¯2
¯
¯
−(τ + τ1 ) α
− xτ
¯W
2
[W f ](τ )e
Wµ,ν (xτ )e
τ dτ ¯¯ =
¯
0
+∞
+∞
¯Z
µZ
0
¯
xτ
− xτ
0
¯
2
=¯
e
Wµ,ν (xτ )
[W f ](τ )e− 2 Wµ,ν (xτ )×
0
0
×e
¯2
¶
¯
1
−(x+ x
) α
τ dτ e
x dx¯¯ ≤
−(τ + τ1 ) α
66
L. P. Castro and M. M. Rodrigues
+∞µ
¶2
Z
0
1
− xτ
0 −(x+ x
) α
2
≤
e
Wµ,ν (xτ )e
x
×
0
+∞
µZ
¶2
− xτ
−(τ + τ1 ) α
2
×
[W f ](τ )e
Wµ,ν (xτ )e
τ dτ dx ≤
0
≤
∗
Cµ,ν
kW f k 2 + −(τ + τ1 ) α
L (R ,e
τ dτ )
+∞
Z
¡
e−
xτ 0
2
¢2
Wµ,ν (xτ 0 ) x2α x−α−1 dx ≤
0
∗
≤ (Cµ,ν
)2 (τ 0 )−α kW f k
1
L2 (R+ ,e−(τ + τ ) τ α dτ )
.
(3.8)
Therefore, for f ∈ HK , we have
+∞
µZ
¶
¡
¢
0
1
−(τ + τ1 ) α
− xτ
2
W
Wµ,ν (xτ )e
τ dτ ∈ L2 R+ , e−(τ + τ 0 ) (τ 0 )α dτ 0
[W f ](τ )e
0
i.e.,
+∞
+∞
µZ
¶
Z
0
xτ
1
1
− xτ
0
[W f ](τ )e− 2 Wµ,ν (xτ )e−(τ + τ ) τ α dτ e−(x+ x ) xα dx
e 2 Wµ,ν (xτ )
0
0
¡
¢
0
1
∈ L2 R+ , e−(τ + τ 0 ) (τ 0 )α dτ 0 .
Indeed, from (3.8), we get
+∞¯·
+∞
¯2
¶¸
µZ
Z
¯
¯
−(τ + τ1 ) α
0 ¯
− xτ
¯ W
2 W
(xτ
)e
τ
dτ
(τ
)
[W
f
](τ
)e
µ,ν
¯
¯ ×
0
0
× e−(τ
≤
0
+ τ10 )
(τ 0 )α dτ 0 ≤
+∞
Z
0
1
e−(τ + τ 0 ) (τ 0 )α (τ 0 )−α dτ 0 ≤
∗
)2 kW f k 2 + −(τ + τ1 ) α
(Cµ,ν
L (R ,e
τ dτ )
0
∗
≤ (Cµ,ν
)2 kW f k
1
L2 (R+ ,e−(τ + τ ) τ α
dτ )
.
¤
∗
Having in mind Lemma 3.2, we are now in a position to define HK
(R+ ) as
+
the space of elements f ∈ HK (R ) which admit the integral representation
+∞
Z
xτ
1
f (x) =
[W f ](τ )e− 2 Wµ,ν (xτ )e−(τ + τ ) τ α dτ.
(3.9)
0
We will now exhibit a significative result based on the representation of
∗
the elements of the space HK
(R+ ) and the definition of the heat kernel.
67
The Weierstrass–Whittaker Integral Transform
∗
Lemma 3.3. Let Kt ∈ HK
(R+ ). Then, the Whittaker type transform
(1.1) of the heat kernel is given by
[W Kt ](τ, x) = e−4ν
2
τ t − xτ
2
e
Wµ,ν (xτ ).
(3.10)
Proof. From Proposition 3.1, we find that Kt ∈ HK (R+ ). Taking into
∗
account the definition of heat kernel (3.3) and since Kt ∈ HK
(R+ ), we get
2
xτ
[W Kt ](τ, x) = e−4ν τ t e− 2 Wµ,ν (xτ ).
¤
4. Properties of the Weierstrass–Whittaker Transform
In this section, we shall define the above-mentioned Weierstrass–Whittaker transform in a formal way, and derive some of its properties.
Definition 4.1. The Weierstrass transform associated with the Whittaker integral transform and called Weierstrass–Whittaker transform, is de1
fined in L2 (R+ , e−(y+ y ) y α dy) by
+∞
Z
1
[Wt f ](x) =
Kt (x, y)f (y)e−(y+ y ) y α dy.
(4.1)
0
For the classical Weierstrass transform, one can see [9].
Proposition 4.2. Let α > max{0, 2ν −2}. For all t > 0, the Weierstrass
1
type transform Wt f is a bounded operator from L2 (R+ , e−(y+ y ) y α dy) into
1
1
L2 (R+ , e−(x+ x ) xα dx) and, for all f ∈ L2 (R+ , e−(y+ y ) y α dy), we have
kWt f k2 2
1
L (R+ ,e−(x+ x ) xα dx)
≤
³
1 ´2
∗
kf k2
.
≤ (Cµ,ν
)2 Γ(0, 1) +
−(y+ 1 ) α
y y dy)
e
L2 (R+ ,e
(4.2)
Proof. The absolutely convergence of the integral (4.1) follows from the
Cauchy–Schwarz inequality and Proposition 3.1. Indeed,
¯
¯
¯[Wt f ](x)¯ ≤
+∞
Z
1
|Kt (x, y)| |f (y)|e−(y+ y ) y α dy ≤
0
+∞
+∞
µZ
¶1/2 µ Z
¶1/2
1
1
≤
|Kt (x, y)|2 e−(y+ y ) y α dy
|f (y)|2 e−(y+ y ) y α dy
≤
0
0
+∞
µZ
¶1/2
∗
2 −(α+1) −(α+1) −(y+ y1 ) α
≤
(Cµ,ν ) x
y
e
y dy
kf k
L2 (R+ ,e
−(y+ 1 ) α
y y
dy)
≤
0
³
1 ´ 12 − α+1
∗
≤ Cµ,ν
Γ(0, 1) +
x 2 kf k 2 + −(y+ y1 ) α .
L (R ,e
y dy)
e
(4.3)
68
L. P. Castro and M. M. Rodrigues
1
Then, for all f ∈ L2 (R+ , e−(y+ y ) y α dy) and using the relation (4.3), we
have
kWt f k2 2
1
L (R+ ,e−(x+ x ) xα dx)
=
+∞
Z
1
|[Wt f ](x)|2 e−(x+ x ) xα dx ≤
0
≤
∗
(Cµ,ν
)2
³
1´
Γ(0, 1) +
kf k2
−(y+ 1 ) α
y y dy)
e
L2 (R+ ,e
∗
≤ (Cµ,ν
)2
³
+∞
Z
1
x−(α+1) e−(x+ x ) xα dx ≤
0
1 ´2
Γ(0, 1) +
kf k2
. ¤
−(y+ 1 ) α
y y dy)
e
L2 (R+ ,e
Proposition 4.3. Let α > max{0, 2ν−2}. For all t > 0, the Weierstrass–
Whittaker transform Wt f belongs to the space HK (R+ ).
Proof. From the previous proposition we have
¡
¢
1
Wt f ∈ L2 R+ , e−(x+ x ) xα dx .
Now, in order to prove that Wt f belongs to the space HK (R+ ), we need to
1
show that W [Wt f ] ∈ L2 (R+ , e−(τ + τ ) τ α dτ ).
From the definition of the Whittaker type transform, we obtain
¯£
¤ ¯
¯ W [Wt f ] (τ )¯ ≤
+∞
Z
xτ
1
e− 2 |Wµ,ν (xτ )| |Wt f (x)|e−(x+ x ) xα dx
0
and by using (4.3) and taking into account the Cauchy–Schwarz inequality,
we have
´1
¯ £
¤ ¯ ³
∗
¯[W Wt f ] (τ )¯ ≤ Γ(0, 1) + 1 2 Cµ,ν
kf k 2 + −(y+ y1 ) α ×
L (R ,e
y dy)
e
+∞
Z
α+1
1
xτ
×
e− 2 |Wµ,ν (xτ )|x− 2 e−(x+ x ) xα dx ≤
0
³
1 ´ 12 ∗
≤ Γ(0, 1) +
Cµ,ν kf k 2 + −(y+ y1 ) α ×
L (R ,e
y dy)
e
+∞
µZ
¶1/2
¡ − xτ
¢2 −(x+ 1 ) α
2
x
×
e
Wµ,ν (xτ ) e
x dx
×
0
+∞
µZ
¶1/2
1
×
x−(α+1) e−(x+ x ) xα dx
≤
0
≤τ
− α+1
2
³
1´ ∗ 3
Γ(0, 1) +
(Cµ,ν ) 2 kf k 2 + −(y+ y1 ) α .
L (R ,e
y dy)
e
69
The Weierstrass–Whittaker Integral Transform
Having in mind the previous inequality, we obtain the following estimate:
°
°
°W [Wt f ]°2 2 + −(τ + 1 ) α
=
τ τ dτ )
L (R ,e
+∞
Z
¯
¯
¯W [Wt f ](τ )¯2 e−(τ + τ1 ) τ α dτ ≤
0
+∞
Z
³
1
1 ´2 ∗ 3
≤ Γ(0, 1) +
(Cµ,ν ) kf k2
τ −(α+1) e−(τ + τ ) τ α dτ ≤
−(y+ 1 ) α
y y dy)
e
L2 (R+ ,e
³
1 ´3 ∗ 3
.
≤ Γ(0, 1) +
(Cµ,ν ) kf k2
−(y+ 1 ) α
y y dy)
e
L2 (R+ ,e
0
(4.4)
Hence, it follows that the composition of the Whittaker type transform
(1.1) with the Weierstrass–Whittaker transform (4.1) belongs to the space
1
L2 (R+ , e−(τ + τ ) τ α dτ ) and therefore Wt f ∈ HK (R+ ).
¤
The just used composition of integral transformations can be described
in an even more detailed way if we invoke the representation of the elements
∗
of the space HK
(R+ ) and the definition of the Weierstrass–Whittaker transform, as we shall see in the next result.
∗
Lemma 4.4. Let Wt f ∈ HK
(R+ ). For all t > 0, we have
£
¤
2
W [Wt f ] (τ ) = e−4ν τ t [W f ](τ ).
(4.5)
Proof. From the definition of Weierstrass–Whittaker transform, the definition of inner product in HK (R+ ), Proposition 3.1, Proposition 4.3 and
Lemma 3.3, we deduce
+∞
Z
1
[Wt f ](x) =
Kt (x, y)f (y)e−(y+ y ) y α dy =
0
+∞
Z
1
=
[W Kt ](τ )W [f ](τ )e−(τ + τ ) τ α dτ =
0
+∞
Z
2
xτ
1
=
e−4ν τ t e− 2 Wµ,ν (xτ )[W f ](τ )e−(τ + τ ) τ α dτ.
0
Since Wt f ∈
∗
HK
(R+ ),
invoking (3.9), we find
£
¤
2
W [Wt f ] (τ ) = e−4ν τ t [W f ](τ ).
(4.6)
¤
5. The Weierstrass–Whittaker Transform as a Solution
of a Heat Type Equation
In this last section we will show that the Weierstrass–Whittaker transform Wt f solves a non-stationary heat type equation (cf. (5.2)). To this
70
L. P. Castro and M. M. Rodrigues
end, first of all, we need to prove that the kernel Kt (x, y) is a solution of a
variant of the heat equation.
We start by recalling that the Whittaker function is an eigenfunction of
a second order differential operator. More precisely,
Az Wµ,ν (z) = 4ν 2 Wµ,ν (z),
where
Az = 4z 2
d2
− z 2 + 4µz + 1.
dz 2
(5.1)
From the differential properties of the Whittaker function, the absolute
and uniform convergence of the integral (1.3) and its derivatives with respect
to t and x, we directly arrive at the following result.
Corollary 5.1. The kernel Kt (x, y) satisfies the non-stationary heat type
equation
∂t u(t, x, y) = −Lx u(t, x, y), t, x, y > 0,
(5.2)
where
Lx = 4τ 3 x2
d2
d
+ 4τ 4 x2
+ τ 3 x2 (τ 2 − 1) + 4µτ 2 x + τ.
2
dx
dx
is a second order differential operator which satisfies
¡ xτ
¢
xτ
Lx e− 2 Wµ,ν (xτ ) = 4ν 2 τ e− 2 Wµ,ν (xτ ).
(5.3)
(5.4)
Furthermore, the kernel Kt (x, y) is also a solution of the non-stationary
heat type equation
∂t u(t, x, y) = −Ly u(t, x, y), t, x, y > 0,
(5.5)
where
Ly = 4τ 3 y 2
d2
d
+ 4τ 4 y 2
+ τ 3 y 2 (τ 2 − 1) + 4µτ 2 y + τ
2
dy
dy
is a second order differential operator which satisfies
¡ yτ
¢
yτ
Ly e− 2 Wµ,ν (yτ ) = 4ν 2 τ e− 2 Wµ,ν (yτ ).
(5.6)
(5.7)
Theorem 5.2. Let f ∈ HK (R+ ). For all t > 0 and for all Wt f ∈
the function Wt f solves the generalized heat equation (5.2), with
the initial condition lim [Wt f ](x) = f (x) in HK (R+ ).
∗
HK
(R+ ),
t→0
Proof. Propositions 3.1 and 4.2 guarantee the necessary differential properties of Wt f , and from the differential properties of the Whittaker function
we deduce that the function Wt f is a solution of (5.2).
The Weierstrass–Whittaker Integral Transform
71
We will now prove the initial condition. From the definition of the norm
of HK (R+ ) (cf. (3.2)) and using Lemma 4.4, we have
kWt f − f k2 2
1
L (R+ ,e−(x+ x ) xα dx)
=
=
+∞
Z
¯£
¯2
¤
1
¯
¯
¯ W [Wt f ] (τ ) − [W f ](τ )¯ e−(τ + τ ) τ α dτ =
0
+∞
Z
¯2
¯ −4ν 2 τ t
¯2 ¯
1
¯e
=
− 1¯ ¯[W f ](τ )¯ e−(τ + τ ) τ α dτ.
(5.8)
0
Since 4ν 2 τ t > 0, we realize that the right-hand side of (5.8) is estimated by
+∞
R
1
|[W f ](τ )|2 e−(τ + τ ) τ α dτ . Then, we can pass to the limit → 0 through
0
equation (5.8) and the desired result is obtained.
¤
Acknowledgments
This work was supported in part by FEDER funds through COMPETE–
Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the
Center for Research and Development in Mathematics and Applications and
the Portuguese Foundation for Science and Technology (“FCT–Fundação
para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011
with COMPETE number FCOMP-01-0124-FEDER-022690.
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(Received 17.09.2013)
Authors’ address:
Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.
E-mail: castro@ua.pt; mrodrigues@ua.pt