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The Inverse Laplace Transform and Analytic Pseudo-Differential Operators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
228, 16]36, 1998
AY986083
The Inverse Laplace Transform and Analytic Pseudo-Differential Operators A. Boumenir Department of Mathematics, Sultan Qaboos Uni¨ ersity, P.O. Box 36, Alkhod, 123 Muscat Oman E-mail: [email protected]
and A. Al-Shuaibi Department of Mathematical Sciences, King Fahd Uni¨ ersity of Petroleum and Minerals, Dhahran, Saudi Arabia E-mail: [email protected] Submitted by H. M. Sri¨ asta¨ a Received November 3, 1997
By comparing the Laplace transform L with the differential operator D, we obtain a formula for the inverse Laplace transform Ly1 s Ž1rp .Vy1 cosŽp D .V L , where V is a unitary transformation operator. This helps us obtain an explicit spectral representation of L . Some applications of the above relation are dis cussed. Q 1998 Academic Press Key Words: Laplace transform; inverse Laplace transform; pseudo-differential operators; differential operator of infinite order
1. INTRODUCTION An extensive bibliography up to 1975 has been compiled by Piessen w18x. Bellman’s book w5x shows a wide range of applications of numerical inversion. Krylov and Skoblya w15x cover the theoretical basis of a number of inversion methods but do not consider implementation or present numerical results. The survey paper by Davies and Martin w8x, which tests 14 inversion procedures on a set of 16 transforms, is a major contribution. However, very few authors have approached the inversion of the Laplace transform from an operator-theoretic point of view by looking at its spectrum and spectral measure Žsee w14x, w19x, and w3x.. This approach, 16 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
PSEUDO-DIFFERENTIAL OPERATORS
17
which is going to be our main concern, will enable us to discover interesting relations between the Laplace transform and differential operators of infinite order. This connection is in fact obtained as a by-product of the spectral measure of the Laplace transform, which is the heart of regularization methods w4x. Although the Laplace transform has been studied for a long time, it still reveals and illustrates deep results in operator and spectral theory. These ideas will be explained as we proceed further. In what follows, the Laplace transform is seen as an operator acting in the separable Hilbert space L2d x Ž0, `., L
L2d x Ž 0, ` . ª L2d x Ž 0, ` . , and so is defined in the L2 sense by L Ž y . Ž s. s
`
ys x
H0 e
y Ž x . dx.
It is well known Žsee w9x. that the Laplace transform of a function of L2d x Ž0, `. belongs to the Hardy space H 2 Žsee w12x.: `
H < F Ž s q it . < s G0 y`
½
H 2 s F Ž s q it . analytic for s ) 0 and sup
2
dt - `
5
L Ž y . Ž s . g H 2 m y g L2d x Ž 0, ` . , and thus the domain of Ly1 coincides with those traces of functions in H 2 on the positive real line that are square integrable. Operators acting in the spaces of analytic functions have a very interesting feature. Indeed, bounded operators acting on the space of entire functions can be represented by differential operators of infinite order Žsee w16x and w11x.. One way of applying or verifying this basic result for the inverse Laplace transform is to use methods of spectral theory. It is readily seen that L is an integral operator with a symmetric kernel, and its square is L 2 Ž y . Ž s. s
`
H0
yŽ x. sqx
dx
where y g L2d x Ž 0, ` . .
The kernel suggests a convolution operator, and this is a crucial link with analytic pseudo-differential operators. To bring in differential operators, we recall that yi drdx is a self-adjoint operator acting in L2d x Žy`, `.. Thus the space L2d x Ž0, `. must be replaced by L2d x Žy`, `., and this is achieved by the following unitary transformation operator V: V : L2 Ž 0, ` . ª L2 Ž y`, ` . ,
18
BOUMENIR AND AL-SHUAIBI
defined by Vy Ž x . s e x r2 y Ž e x .
and Vy1 g Ž x . s
1
'x g Ž ln x . .
It is readily seen that Vy1 s V *,
VV * s I, and so where the adjoint V * is defined by `
`
Hy`Vf Ž t . c Ž t . dt s H0
f Ž t . V *c Ž t . dt.
We now define an operator A acting in L2d x Žy`, `., L2d x Ž y`, ` .
6
A
L2d x Ž y`, ` . Vy1 x
V L2
Ž 0, ` .
L2d x
6
L2d x
Ž 1.1.
Ž 0, ` .
A [ V L 2 Vy1 . It is easily verified that A is an integral operator of the Carleman type defined by a convolution Ay Ž x . s k ) y Ž x . s
`
Hy`k Ž x y h . y Ž h . dh ,
Ž 1.2.
where kŽ x. s
1 2 cosh Ž xr2.
.
We shall agree to denote the Fourier transform as an operator acting in n
L2d l Ž y`, ` .
6
L2d x Ž y`, ` . by fˆŽ l . [
`
Hy` f Ž x . e
il x
dx and
f Ž x. [
`
yi l x
Hy` fˆŽ l. e
d
l 2p
.
We recall that Fourier transform is first defined on the set of continuous functions with compact support and then is extended by continuity to L2d x Žy`, `., that is by closing its graph. Thus we obtain a unitary operator, and this fact is known as the Parseval equality for f, c g L2d x Žy`, `., `
`
l
y`
y`
2p
Ž f , c . s H f Ž x . c Ž x . dx s H fˆŽ l . cˆ Ž l . d
.
PSEUDO-DIFFERENTIAL OPERATORS
19
PROPOSITION 1.1. The operator A defined on L2d x Žy`, `. by Ž1.2. is a bounded self-adjoint operator with 5 A 5 s p . Proof. It is easy to see that the Fourier transform
p
ˆk Ž l . s
cosh pl
is a bounded function. Using the Parseval relation and the fact that suplg R < ˆ k Ž l.< F p , we have for f g L2d x Žy`, `. `
Hy`< Af Ž x . <
2
dx s s
`
$
l
Hy`< AfŽ l. < d 2p 2
`
l
Hy`< ˆk Ž l. fˆŽ l. < d 2p 2
F sup < ˆ k Ž l. < 2 lg R
`
l
Hy`< fˆŽ l. < d 2p
`
2
`
2
2
l
Fp2
Hy`< fˆŽ l. < d 2p
Fp2
Hy`< f Ž x . <
dx.
Hence DA s L2d x Žy`, `., 5 A 5 F p , and in fact the equality holds. Since A is bounded and symmetric, it is a self-adjoint operator in L2d x Žy`, `. Žsee w1x.. COROLLARY 1.2. The operator L 2 is a positi¨ e bounded self-adjoint operator acting in L2d x Ž0, `. and, 5 L 25 s p. Proof. It follows from Ž1.1. that L 2 is unitarily equivalent to A, which is a bounded self-adjoint operator. 2. THE SPECTRAL REPRESENTATION OF L 2 To find the spectral function of L 2 , we only need to find the spectral function of A. By using the Parseval relation where f, c g L2d x Žy`, `., we
20
BOUMENIR AND AL-SHUAIBI
have `
Ž Af , c . s H Af Ž x . c Ž x . dx y` `
s
Hy`k ) f Ž x . c Ž x . dx
s
Hy`k ) f Ž l. cˆ Ž l. d 2p
s
Hy`k ) f Ž l. cˆ Ž l. d 2p
s
Hy` fˆŽ l. cˆ Ž l. ˆk Ž l. d 2p .
`
$
l
`
$
l
l
`
Ž 2.1.
Let a be the increasing function defined by
ˆk Ž aŽ l . . s
p cosh p a Ž l .
s l,
a Ž l . - 0 for 0 - l - p ,
and so a is defined by
aŽ l . [
y1
p
ln
p l
q
(
p2 l2
y1 .
Ž 2.2.
We have from Ž2.1. p
Ž Af , c . s H l fˆŽ aŽ l . . ? cˆ Ž aŽ l . . d 0
aŽ l. 2p
0
Hp l fˆŽ yaŽ l. . cˆ Ž yaŽ l. . d
q s
yaŽ l . 2p
p
H0 l fˆŽ aŽ l. . ? cˆ Ž aŽ l. . 4 q fˆŽ yaŽ l . . ? cˆ Ž yaŽ l . . 4 d
aŽ l. 2p
.
PSEUDO-DIFFERENTIAL OPERATORS
21
The above Parseval equation can then be written in a matrix form as 1
p
Ž Af , c . sH l fˆŽ aŽ l . . , fˆŽ yaŽ l . . 0
2p
d
aŽ l. 0
0 aŽ l.
cˆ Ž a Ž l . . cˆ Ž yaŽ l . .
.
Ž 2.3. This only means that e i x aŽ l ., eyi x aŽ l.4 are the eigenfunctionals of A, the multiplicity of the spectrum is 2, and the associated spectral matrix is aŽ l . 0 2p 1
0 . aŽ l.
PROPOSITION 2.1. The spectral matrix of A is aŽ l . 0 2p 1
0 , aŽ l.
and the multiplicity of the spectrum s s supp daŽ l. s w0, p x is 2. Remark. The spectrum of A is continuous, and therefore the ‘‘eigenfunctions’’ are distributions, namely, e " i x aŽ l . that are outside L2d x Žy`, `., 5 e " i x aŽ l . 5 s `. Thus they are called eigenfunctionals or generalized functions Žsee w13, Vol. 3x.. Now we use our relation A s V L 2 Vy1 to derive the spectral representation for L 2 . We shall assume that f Ž x ., c Ž x . g C0 Ž0, `., where C0 Ž0, `. is the space of continuous functions with compact support, and so integrals are understood in the distributional sense, `
H0
L 2 f Ž x . c Ž x . dx s
p
H0 l f˜Ž l.
t
dL Ž l . f˜Ž l . ,
Ž 2.4.
where L is the spectral matrix associated with L 2 and f˜Ž l. is its transform. On the other hand, using Parseval equality, Ž2.3. leads to `
H0
L 2 f Ž x . c Ž x . dx s Ž AVf , Vc . s
p
H0 l Vfˆ Ž aŽ l. . , Vfˆ Ž yaŽ l. . =
aŽ l. 0
0 aŽ l.
1 2p
Vˆc Ž a Ž l . . Vˆc Ž yaŽ l . .
d
.
22
BOUMENIR AND AL-SHUAIBI
This simply means that the transform associated with L 2 is f˜Ž l . s
ˆ Ž aŽ l . . Vf
ž
ˆ Ž yaŽ l . . Vf
/
,
and the spectral matrix is given by L Ž l. s
aŽ l. 0 2p 1
0 . aŽ l.
Thus the spectrum of L 2 is continuous of multiplicity 2 and covers w0, p x. The corresponding eigenfunctionals are f˜Ž l . s
s
s
ž ž ž
ˆ Ž aŽ l. . Vf ˆ Ž yaŽ l . . Vf
/
H0` f Ž x . Vy1 e i x aŽ l. dx H0` f Ž x . Vy1 eyi x aŽ l . dx H0` f Ž x . xy
1 2
qi aŽ l .
dx
H0` f Ž x . xy
1 2
yi aŽ l .
dx
/
/
,
and thus the eigenfunctionals are xy1 r2qi aŽ l ., xy1r2yi aŽ l.4 . Let us denote the components of f˜Ž l. by `
Tq Ž f . Ž l . s
H0
Ty Ž f . Ž l . s
H0
`
f Ž x . xy1 r2qi aŽ l . dx f Ž x . xy1 r2yi aŽ l . dx.
Remark. Since f has a compact support, the integrals defining f˜ are well defined.
3. THE SPECTRAL RESOLUTION OF L It is readily seen that L is the square root of L 2 . For simplicity let us denote w Ž l . [ y 12 q iaŽ l2 . ,
l g Ž y'p , 'p . ,
PSEUDO-DIFFERENTIAL OPERATORS
23
and the classical Euler]Gamma function by G Ž x .. With the help of the eigenfunctionals of L 2 , we can reconstruct the eigenfunctionals of L . This is achieved with transmutations or what is called a transformation operator, a very useful device in inverse spectral problems Žsee w16x and w6x.. It is easy to see that the newly defined functions y Ž x, l . s G Ž 1 q w Ž l . . x wŽ l. q sign Ž l . < G Ž 1 q w Ž l . . < x wŽ l.
Ž 3.1.
satisfy L y s l y, where sign Ž l . s
½
q1 y1
l)0 l - 0,
and recall that l s signŽ l.< G Ž1 q w Ž l..<. Now use y to define the transform associated with L by ^
f Ž l. s
`
H0
f Ž x . y Ž x, l . dx
f g C0 Ž 0, ` . ,
and thus for l g Žy 'p , 'p ., ^
f Ž l . sG Ž 1qw Ž l . . ?Ty Ž f . Ž l2 . qsign Ž l . ? < G Ž 1qw Ž l . . < Tq Ž f . Ž l2 . .
Ž 3.2. It is clear that the multiplicity of the spectrum of L is either 1 or 2. We claim that Eq. Ž3.1. defines a complete system of eigenfunctionals, and so the multiplicity is only 1. Recall that the system of eigenfunctionals y Ž x, l. is complete if ^
f Ž l . s 0 for
l g Ž y'p , 'p . « f s 0
in L2d x Ž 0, ` . .
^
Using Ž3.2. for l ) 0 and l - 0, condition f Ž l. s 0 is equivalent to the following system: G Ž 1 q w . ? Ty Ž f . Ž l 2 . q < G Ž 1 q w . < ? Tq Ž f . Ž l 2 . s 0 G Ž 1 q w . ? Ty Ž f . Ž l2 . y < G Ž 1 q w . < ? Tq Ž f . Ž l2 . s 0. Since its determinant is nonzero, we have
Ž Ty Ž f . Ž l2 . , Tq Ž f . Ž l2 . . s f˜Ž l2 . s 0, that is, f s 0 in L2d x Ž0, `.. Hence the set y Ž x, l. for l g Žy 'p , 'p .4 is complete in L2d x Ž0, `..
24
BOUMENIR AND AL-SHUAIBI
We now deduce a spectral representation for L from Ž2.4.. The transition formula Ž3.2., together with the Parseval relation, implies for f, c g L2d x Ž0, `., since the domain of L 2 and L coincides with L2d x Ž0, `., that
Ž L f , Lc . s Ž L 2 f , c . . Upon using the Parseval equality associated with the operator L and L 2 , we obtain
Hy''pp l f Ž l. c Ž l. d r Ž l. ^
2
^
s
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
,
where r Ž l. is the spectral function associated with L , which is to be determined here. Recall that < G Ž1 q w .< 2 s l2 , and use Ž3.2. to obtain
Hy''pp l f Ž l. c Ž l. d r Ž l. 2
s
^
^
Hy''pp l G Ž 1 q w Ž l. . ? T 2
y
Ž f . Ž l2 .
qsign Ž l . ? < G Ž 1 q w Ž l . . < Tq Ž f . Ž l2 . 4 ? ? GŽ 1qwŽ l. . ?Ty Ž c. Ž l2. qsign Ž l. ?< G Ž 1qw Ž l. . < Tq Ž c. Ž l2. 4 d r Ž l . s
Hy''pp l T 4
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
Hy''pp signŽ l. ? G Ž w q 1. < G Ž 1 q w . < T
q
y
Ž f . Ž l 2 . Tq Ž c . Ž l 2 .
qG Ž 1 q w . < G Ž 1 q w . < Ty Ž f . Ž l2 . Tq Ž c . Ž l2 . 4 d r Ž l . . Since the second integrand is an odd function, we only need to assume that r Ž l. is an odd function for the above expression to reduce to
Hy''pp l f Ž l. c Ž l. d r Ž l. ^
2
s
^
Hy''pp l T 4
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
PSEUDO-DIFFERENTIAL OPERATORS
25
to obtain
Hy''pp l T 4
s
y
Ž f . Ž l 2 . T y Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
.
Therefore we have
H0'p l t
2
4
s
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q T q Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
.
Since the above relation holds for all f and c in C0 Ž0, `., which is dense in L2d x Ž0, `., the measures must be equal: 2 l2 d r Ž 'l . s s s s s s
l 2p
daŽ l .
0-l-p
l daŽ l . 2p
dl
dl
l 2pl'p 2 y l2
l
dl dl
2pl'p y l 2
1 2p'p 2 y l2
'l p'p 2 y l2
2
d'l
2'l d'l d'l ,
and hence
r 9 Ž 'l . s r 9 Ž l. s
d'l
'l 2pl2'p 2 y l2 1 2pl3'p 2 y l4
.
26
BOUMENIR AND AL-SHUAIBI
Thus we obtain
r 9 Ž l. s
1 2pl3'p 2 y l4
,
where 0 F l F 'p ,
and to recover r Ž l. over all of s , we need to recall that r Ž l. is an odd function,
r 9 Ž l. s
PROPOSITION 3.1. f Ž x. s
1 2p < l < 3'p 2 y l4
y'p F l F 'p .
Ž 3.3.
If f g C0 Ž0, `., then we ha¨ e 1
Hy''pp f Ž l. y Ž x, l. 2p < l< 'p ^
3
2
y l4
dl,
and F g H 2 l L2d x Ž0, `., L y1 F Ž x . s
1
Hy''pp F Ž l. y Ž x, l. 2p < l< l'p ^
3
2
y l4
dl.
Ž 3.4.
Recall that the spectral representations is not unique. A simple rearrangement will allow us to obtain an increasing spectral function. Indeed, we can define
m Ž l. [
½
rŽ l y p . rŽ l q p .
for l G 0 for l - 0.
Equation Ž3.4. opens the possibility of using regularization methods based on the spectral properties such as those found in w4x. It is clear that the spectral function has three points to regularize,
l s y'p , 0, and 'p .
4. ANALYTIC PSEUDO-DIFFERENTIAL OPERATORS The following theorem furnishes a real inversion formula for L y and unveils the nature of the unboundedness of the inverse Laplace transform operator. 1
PSEUDO-DIFFERENTIAL OPERATORS
THEOREM 4.1.
Let F g H 2 l L2d x Ž0, `.. Then
Ly F s 1
27
1
p
Vy1 cos Ž p D . V L F
where D [
d dx
.
Proof. It is known that convolution operators can be represented as differential operators. Let f g L2d x Žy`, `. be such that fˆ has a compact support; then the Fourier transform applied to Ž1.2. yields Af Ž x . ' k ) f Ž x . $
Af Ž l. s ˆ k Ž l . fˆ Ž l . p s fˆ Ž l . cosh pl
$
Ay1 f Ž l . s s
cosh pl
p
fˆ Ž l . $
1
cosh Ž p Ž yiD . . f Ž l . ,
p
where D s drdx. Thus by taking the inverse Fourier transform we have Ay1f Ž x . s
1
p
cos Ž p D . f Ž x . .
Ž 4.1.
Observe that since fˆ has a compact support, f is infinitely many times differentiable. Now that we have a representation for the operator Ay1 , we can deduce from Ž1.1. the inverse Ly2 s Vy1Ay1 V 1 s Vy1 cos Ž p D . V p Ly1 s
1
p
Ž 4.2.
Vy1 cos Ž p D . V L ,
which is valid if V L F is in the domain of cosŽp D .. We now show that F g H 2 l L2d x Ž0, `., i.e., in the range of Laplace transform L , is sufficient for V L F to be in the domain of cosŽp D .. Indeed, if F g H 2 l L2d x Ž0, `., then there exists f g L2d x Ž0, `. such that F s L f, and so VLF s VL 2f.
28
BOUMENIR AND AL-SHUAIBI
Upon using Ž1.1. we obtain V L F s AVf , which means that Vf is the domain of A, and V L F is in the range of A, and, in other words, V L F is in the domain of Ay1 . Hence Ž4.1. implies that V L F is in the domain of Ž1rp .cosŽp D ., i.e., F g H 2 l L2d x Ž 0, ` . «
1
p
cos Ž p D . V L F g L2d x Ž y`, ` . ,
and so Ž4.2. holds in H 2 l L2d x Ž0, `.. Equation Ž4.2. establishes a direct connection of the inverse Laplace transform with pseudo-differential operators, or differential operators of infinite order. This allows us to use many results related to approximation Žsee w11x.. It is easy to see that the ill-posedness of the inverse transform is really due to cosŽp D . only, since all of the remaining operators are bounded in Ž4.2., and hence any regularization should deal with cosŽp D . only. We recall that cosŽp D . is a bounded operator in the space ExpAŽ C . Žsee w10x and w11x., where Expr Ž C . s g Ž z . entire < g Ž z . < F M exp Ž r < z < . 4 . It is readily seen that Expr Ž C . is a Banach space with norm 5 u 5 r [ sup < u Ž z .
Recall that if r 1 -r 2 ; then we have the compact embedding Expr 1Ž C . ¨ Expr 2Ž C .. These compact embeddings help the construction of countably normed spaces, as an inductive limit, ExpA Ž C . s lim ind Expr Ž C . , r A
F Ž s.
6
Exp AŽ C .
6
and a sequence Fn is said to converge in ExpAŽ C ., i.e., FnŽ s . Expr Ž C . if there exists r - A, such that FnŽ s . F Ž s ., i.e., ;K compact sup < Fn Ž z . y F Ž z .
We now apply the above results to improve the inverse Laplace transform as defined by Ž4.2.. Recall that for an entire function, the Taylor series represents a translation, and so for a function t analytic in a
PSEUDO-DIFFERENTIAL OPERATORS
29
neighborhood of the point a,
t Ž a q h. s
hn
Ý nG0
n!
D nt Ž a .
s exp Ž hD . t Ž a . . Using the above representation we have cos Ž p D . g Ž x . s s
exp Ž ip D . q exp Ž yip D . 2 1 2
gŽ z.
Ž g Ž x q ip . q g Ž x y ip . . ,
Ž 4.3.
which is valid when g in defined on the real line and has an analytic extension in the strip V [
1 2
Ž V L F Ž x q ip . q V L F Ž x y ip . .
to hold. Thus we have THEOREM 4.2. v v
If
F g H l L2d x Ž0, `., V L F Ž x . has an analytic extension in the strip V [ < Im z < F p 4 , then 2
L y1 F Ž x . s Vy1
V L F Ž x q ip . q V L F Ž x y ip . 2p
.
Ž 4.4.
We now simplify formula Ž4.4. by using the Schwartz reflection principle Žsee w7x.. Indeed, observe that if F Ž s . is real for real s, then V L F Ž x . is also real for real x. Hence by the Schwartz reflection principle, applied in the strip V [
30
BOUMENIR AND AL-SHUAIBI
Hence we have proved COROLLARY 4.3. v v v
Let the following conditions hold:
F Ž s . is real for real s. F g H 2 l L2d x Ž0, `.. V L F Ž x . has an analytic extension in the strip V [ < Im z < F p 4 .
Then L y1 F Ž x . s
1
p
Vy1 Re V L F Ž x q ip . .
Ž 4.5.
Remark. In Ž4.4., the operators V and Vy1 do not cancel, as it would mean that cosŽp D . commutes with V.
5. EXAMPLES We now consider examples where F Ž s . is real for real s, and we shall use Ž4.5. to compute the inverse of the Laplace transform F. For simplicity we shall start with f g L2d x Ž0, `., to guarantee that F g H 2 l L2d x Ž0, `., i.e., it is in the range of the Laplace transform. EXAMPLE 1. Consider f Ž t . s Ž1r 't .w H Ž t y 1. y H Ž t y 36.x, which belongs to L2d x Ž0, `. and whose Laplace transform is F Ž s. s L f Ž s. s
'p 's
Er fc Ž 's . y Er fc Ž 6's . ,
where Er fcŽ's . is the error function Er fc Ž x . s 1 y
2
x
'p H0 exp Ž yt
2
. dt.
By construction, F is in the range of L , and so Corollary 4.3 can be applied to obtain f s L y1 F s Ž1rp .Vy1 Rew V L F Ž x q ip .x. Taking the Laplace again, we obtain L FŽ x. s
2
'x
arctan Ž 'x . y arctan
'x
ž / 6
,
PSEUDO-DIFFERENTIAL OPERATORS
31
which is an entire function of x, and so Theorem 4.2 applies: 1
Re V L F Ž x q ip . s
p
2
p
Re arctan y exp Ž x . y arctan
'
(
y1 36
exp Ž x . .
Using the previous equation, we have 1
p
1
Re V L F Ž x q ip . s
2
csign 1 y
ž
1 6
/
'expŽ x .
y csign 1 y exp Ž x . ,
ž
'
/
and so Vy1
1
p
Re V L F Ž t . s
1
csign 1 y
ž
2't
1
't
6
/
y csign Ž 1 y 't . . Ž 5.1.
Then, using the fact that
½
csign Ž t . s
if Re Ž t . G 0 if Re Ž t . - 0
1 y1
s 1 y 2 H Ž yt . , that is for t G 0, csign Ž 1 y 't . s 1 y 2 H Ž 't y 1 . s 1 y 2 H Ž t y 1. , from Ž5.1. we have Vy1
1
p
cos Ž p D . V L F Ž t . s s
1
2 H Ž t y 1 . y 2 H Ž tr36 y 1 .
2't 1
't
H Ž t y 1 . y H Ž t y 36 . .
We obtain the required result in the L2d x Ž0, `. sense Vy1
1
p
cos Ž p D . V L F Ž t . s f Ž t . .
EXAMPLE 2. Let f Ž t. s
½
0 y1rt
tF1 t ) 1,
32
BOUMENIR AND AL-SHUAIBI
which obviously belongs to the domain of L . Its Laplace transform is given by F Ž s . s L Ž s . s Ei Ž ys . , where Ei denotes the exponential integral defined by exp Ž t .
`
Ei Ž x . s PV
Hy`
t
dt.
Its Laplace transform is given by L F Ž s. s
y1
ln Ž 1 q s .
s
and V L F Ž x . s exp
yx
ž / 2
ln Ž 1 q exp Ž x . . .
Clearly, V L F Ž x . is entire in x. Then by the Schwartz reflection principle we have 1
p
cos Ž p D . V L F Ž x . s s
1
p 1
p
Re yI exp exp
yx
ž / 2
yx
ž / 2
ln Ž 1 y exp Ž x . .
arg Ž 1 y exp Ž x . . ,
which leads to Vy1
1
p
cos Ž p D . V L F Ž t . s
1
pt
arg Ž 1 y t . .
Recall that arg Ž j . s
½
0 p
if j ) 0 if j - 0.
Ž 5.2.
Thus we have recovered f in the L2d x Ž0, `. sense, Vy1
1
p
cos Ž p D . V L F Ž t . s
1 t
H Ž t y 1. ,
where H is the Heaviside step function. EXAMPLE 3. Let us take a simple function in L2d x Ž0, `., defined by a pulse f Ž t . s H Ž t y 1. y H Ž t y 2. .
PSEUDO-DIFFERENTIAL OPERATORS
33
Then F Ž s. s L f Ž s. 1
s
s
exp Ž ys . y exp Ž y2 s .
is entire and belongs to the range of L . Next we compute L F Ž x . s ln Ž x q 2 . y ln Ž x q 1 . , and so V L F Ž x . s exp
x
ž / 2
ln Ž 2 q exp Ž x . . y ln Ž 1 q exp Ž x . . .
By the Schwartz reflection principle we have 1
p
cos Ž p D . V L F Ž x . s s
1
p
y1
p
x
ln Ž 2 yexp Ž x . . y ln Ž 1 y exp Ž x . .
½ ž / ž / Ž
Re I exp exp
x
2
2
5
arg 2 y exp Ž x . . y arg Ž 1 y exp Ž x . . .
Thus Vy1
1
p
1
cos Ž p D . V L F Ž t . s
p
arg Ž 1 y t . y arg Ž 2 y t . ,
and upon using Ž5.2. we obtain in the L2d x Ž0, `. sense that Vy1
1
p
cos Ž p D . V L F Ž t . s H Ž t y 1 . y H Ž t y 2 . s f Ž t. .
The next example deals with the transform of the Sinc function. EXAMPLE 4. Let us denote f Ž t. s
sin Ž t . t
,
which obviously belongs to the domain of L . Then its Laplace is given by F Ž s. s
p 2
y arctan Ž s . .
34
BOUMENIR AND AL-SHUAIBI
The second iteration by Laplace Žsee w9x. gives L F Ž s. s
p 2s
y
sin Ž s . s
Ci Ž s . q cos Ž s .
Ssi Ž s . s
.
Transforming the result by V yields p VL FŽ x. s y sin Ž exp Ž x . . Ci Ž exp Ž x . . 2 qcos Ž exp Ž x . . Ssi Ž exp Ž x . . exp y
x
ž / 2
.
Ž 5.3.
We now recall that the sine integral functions Si Ž x . s
x
H0
sin Ž t .
Ssi Ž x . s Si Ž x . y
dt
t
p 2
are real functions for real x, and Ci Ž x . s g q ln Ž x . q
x
H0
cos Ž t . y 1 t
dt
is also real if x ) 0. Thus we deduce that V L F is an entire function in x that is real for real x, and thus the Schwartz reflection principle implies that 1
p
cos Ž p D . V L F Ž x . s
1
p
Re yI exp y
½
x
p
2
2
ž /
y sin Ž yexp Ž x . . Ci Ž yexp Ž x . .
qcos Ž yexp Ž x . . Ssi Ž yexp Ž x . . s
1
p
exp y
x
ž / 2
Im
p 2
q sin Ž exp Ž x . . Ci Ž yexp Ž x . . qcos Ž exp Ž x . . Ssi Ž yexp Ž x . .
s s
1
exp y
x
sin Ž exp Ž x . . Im Ci Ž yexp Ž x . .
ž / p ž / p ž / Ž 1
exp y
s exp y
x
2
2 x
2
sin Ž exp Ž x . . Im ln Ž yexp Ž x . .
sin exp Ž x . . .
5
PSEUDO-DIFFERENTIAL OPERATORS
35
Applying Vy1 leads us to Vy1
1
p
cos Ž p D . V L F Ž t . s
sin Ž t . t
,
which holds in the L2d x Ž0, `. sense. EXAMPLE 5. Consider f Ž t . s 3t 2 expŽyt .; then its Laplace transform is F Ž s. s
6
Ž s q 1.
3
,
and L F Ž s . s 3 y 3s q 3s 2 exp Ž s . Ei Ž 1, s . . Applying the V operator yields V L F Ž x . s 3 exp Ž x . 1 y exp Ž x . q Ei Ž 1, exp Ž x . . exp Ž exp Ž x . q 2 x .
'
and 1
p
cos Ž p D . V L F Ž x . s 3 exp
5
x exp Ž yexp Ž x . . ,
ž / 2
which implies in the L2d x Ž0, `. sense that Vy1
1
p
cos Ž p D . V L F Ž t . s 3t 2 exp Ž yt . .
6. CONCLUSION Using operational calculus and pseudo-differential operators, we have reduced the inverse Laplace transform to simple operations. Indeed, all we need to do is to take the Laplace transform L F Ž s . where F Ž s . is the given transform and then use translation operators with a change of variable. One should recall that similar expressions were obtained in w19x, except for a complicated symbol involving the inverse of the Euler]Gamma function, which remained a major obstacle for simple computational applications. This difficulty is removed in our case, since the symbol cosŽp D . is entire and can be expressed in terms of simple translation operators.
ACKNOWLEDGMENT The authors gratefully acknowledge the support provided by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.
36
BOUMENIR AND AL-SHUAIBI
REFERENCES 1. D. N. Akhieser and I. Glazman, ‘‘Theory of Linear Operators in Hilbert Spaces,’’ Dover, New York, 1963. 2. D. Ang, J. Lund, and F. Stenger, Complex variable and regularization methods of inversion of the Laplace transform, Math. Comp. 53 Ž1988., 589]609. 3. A. Alshuaibi, On the spectral measure of the Laplace transform operator, Arabian J. Sci. Engrg. 22 Ž1A. Ž1997., 87]97. 4. A. B. Bakuskii, Regularized algorithms for linear equations with unbounded operators, So¨ iet Math. Dokl. 9 Ž6. Ž1968., 1298]1300. 5. R. Bellman, R. E. Kalaba, and J. A. Lockett, ‘‘Numerical Inversion of the Laplace Transform,’’ Elsevier, New York, 1966. 6. A. Boumenir, A comparison theorem for self-adjoint operators, Proc. Amer. Math. Soc. 111 Ž1. Ž1991., 161]175. 7. J. Connay, ‘‘Functions of One Complex Variable,’’ Springer Verlag, BerlinrNew York, Graduate Texts in Mathematics 11, 1984. 8. B. Davies and B. Martin, Numerical inversion of the Laplace transform: A survey and comparison of methods, J. Comp. Phys. 33 Ž1979., 1]32. 9. V. Ditkin and A. Proudnikov, ‘‘Transformations Integrales et Calcul Operationel,’’ MIR, Moscow, 1978. 10. J. A. Dubinski, Analytic pseudo-differential operators and their applications, Math. Appl. 68 Ž1991.. 11. T. Duc Van and D. Nho Hao, ‘‘Differential Operators of Infinite Order with Real Argument and Their Applications,’’ World Scientific, Singapore, 1994. 12. K. Hoffman, ‘‘Banach Spaces of Analytic Functions,’’ Dover, New York, 1962. 13. I. M. Gelfand and G. Shilov, ‘‘Generalized Functions,’’ Academic Press, San Diego, 1964. 14. D. S. Gilliam, J. R. Schulenberger, and J. R. Lund, Spectral representation of the Laplace and Stieljes transform, in ‘‘Computational and Combinatorial Methods in System Theory’’ ŽC. P. Byrne and Lindquist, Eds.., pp. 69]74. Elsevier Science, North-Holland, Amsterdam, 1986. 15. V. I. Krylov and N. S. Skoblya, ‘‘A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transform,’’ Mir, Moscow, 1977. 16. B. M. Levitan, Lie theorems for generalized translations, Amer. Math. Soc. Transl. Ser. 2, 105 Ž1976., 211]220. 17. J. G. MacWhirter and E. R. Picke, On numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A 11 Ž1978., 729]1745. 18. R. Piessens, A bibliography on numerical inversion of the Laplace transform and application, J. Comput. Appl. Math. 1 Ž1975., 115]128. 19. H. Pollard, Integral transforms, Duke Math. J. 13 Ž1946., 307]330.
228, 16]36, 1998
AY986083
The Inverse Laplace Transform and Analytic Pseudo-Differential Operators A. Boumenir Department of Mathematics, Sultan Qaboos Uni¨ ersity, P.O. Box 36, Alkhod, 123 Muscat Oman E-mail: [email protected]
and A. Al-Shuaibi Department of Mathematical Sciences, King Fahd Uni¨ ersity of Petroleum and Minerals, Dhahran, Saudi Arabia E-mail: [email protected] Submitted by H. M. Sri¨ asta¨ a Received November 3, 1997
By comparing the Laplace transform L with the differential operator D, we obtain a formula for the inverse Laplace transform Ly1 s Ž1rp .Vy1 cosŽp D .V L , where V is a unitary transformation operator. This helps us obtain an explicit spectral representation of L . Some applications of the above relation are dis cussed. Q 1998 Academic Press Key Words: Laplace transform; inverse Laplace transform; pseudo-differential operators; differential operator of infinite order
1. INTRODUCTION An extensive bibliography up to 1975 has been compiled by Piessen w18x. Bellman’s book w5x shows a wide range of applications of numerical inversion. Krylov and Skoblya w15x cover the theoretical basis of a number of inversion methods but do not consider implementation or present numerical results. The survey paper by Davies and Martin w8x, which tests 14 inversion procedures on a set of 16 transforms, is a major contribution. However, very few authors have approached the inversion of the Laplace transform from an operator-theoretic point of view by looking at its spectrum and spectral measure Žsee w14x, w19x, and w3x.. This approach, 16 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
PSEUDO-DIFFERENTIAL OPERATORS
17
which is going to be our main concern, will enable us to discover interesting relations between the Laplace transform and differential operators of infinite order. This connection is in fact obtained as a by-product of the spectral measure of the Laplace transform, which is the heart of regularization methods w4x. Although the Laplace transform has been studied for a long time, it still reveals and illustrates deep results in operator and spectral theory. These ideas will be explained as we proceed further. In what follows, the Laplace transform is seen as an operator acting in the separable Hilbert space L2d x Ž0, `., L
L2d x Ž 0, ` . ª L2d x Ž 0, ` . , and so is defined in the L2 sense by L Ž y . Ž s. s
`
ys x
H0 e
y Ž x . dx.
It is well known Žsee w9x. that the Laplace transform of a function of L2d x Ž0, `. belongs to the Hardy space H 2 Žsee w12x.: `
H < F Ž s q it . < s G0 y`
½
H 2 s F Ž s q it . analytic for s ) 0 and sup
2
dt - `
5
L Ž y . Ž s . g H 2 m y g L2d x Ž 0, ` . , and thus the domain of Ly1 coincides with those traces of functions in H 2 on the positive real line that are square integrable. Operators acting in the spaces of analytic functions have a very interesting feature. Indeed, bounded operators acting on the space of entire functions can be represented by differential operators of infinite order Žsee w16x and w11x.. One way of applying or verifying this basic result for the inverse Laplace transform is to use methods of spectral theory. It is readily seen that L is an integral operator with a symmetric kernel, and its square is L 2 Ž y . Ž s. s
`
H0
yŽ x. sqx
dx
where y g L2d x Ž 0, ` . .
The kernel suggests a convolution operator, and this is a crucial link with analytic pseudo-differential operators. To bring in differential operators, we recall that yi drdx is a self-adjoint operator acting in L2d x Žy`, `.. Thus the space L2d x Ž0, `. must be replaced by L2d x Žy`, `., and this is achieved by the following unitary transformation operator V: V : L2 Ž 0, ` . ª L2 Ž y`, ` . ,
18
BOUMENIR AND AL-SHUAIBI
defined by Vy Ž x . s e x r2 y Ž e x .
and Vy1 g Ž x . s
1
'x g Ž ln x . .
It is readily seen that Vy1 s V *,
VV * s I, and so where the adjoint V * is defined by `
`
Hy`Vf Ž t . c Ž t . dt s H0
f Ž t . V *c Ž t . dt.
We now define an operator A acting in L2d x Žy`, `., L2d x Ž y`, ` .
6
A
L2d x Ž y`, ` . Vy1 x
V L2
Ž 0, ` .
L2d x
6
L2d x
Ž 1.1.
Ž 0, ` .
A [ V L 2 Vy1 . It is easily verified that A is an integral operator of the Carleman type defined by a convolution Ay Ž x . s k ) y Ž x . s
`
Hy`k Ž x y h . y Ž h . dh ,
Ž 1.2.
where kŽ x. s
1 2 cosh Ž xr2.
.
We shall agree to denote the Fourier transform as an operator acting in n
L2d l Ž y`, ` .
6
L2d x Ž y`, ` . by fˆŽ l . [
`
Hy` f Ž x . e
il x
dx and
f Ž x. [
`
yi l x
Hy` fˆŽ l. e
d
l 2p
.
We recall that Fourier transform is first defined on the set of continuous functions with compact support and then is extended by continuity to L2d x Žy`, `., that is by closing its graph. Thus we obtain a unitary operator, and this fact is known as the Parseval equality for f, c g L2d x Žy`, `., `
`
l
y`
y`
2p
Ž f , c . s H f Ž x . c Ž x . dx s H fˆŽ l . cˆ Ž l . d
.
PSEUDO-DIFFERENTIAL OPERATORS
19
PROPOSITION 1.1. The operator A defined on L2d x Žy`, `. by Ž1.2. is a bounded self-adjoint operator with 5 A 5 s p . Proof. It is easy to see that the Fourier transform
p
ˆk Ž l . s
cosh pl
is a bounded function. Using the Parseval relation and the fact that suplg R < ˆ k Ž l.< F p , we have for f g L2d x Žy`, `. `
Hy`< Af Ž x . <
2
dx s s
`
$
l
Hy`< AfŽ l. < d 2p 2
`
l
Hy`< ˆk Ž l. fˆŽ l. < d 2p 2
F sup < ˆ k Ž l. < 2 lg R
`
l
Hy`< fˆŽ l. < d 2p
`
2
`
2
2
l
Fp2
Hy`< fˆŽ l. < d 2p
Fp2
Hy`< f Ž x . <
dx.
Hence DA s L2d x Žy`, `., 5 A 5 F p , and in fact the equality holds. Since A is bounded and symmetric, it is a self-adjoint operator in L2d x Žy`, `. Žsee w1x.. COROLLARY 1.2. The operator L 2 is a positi¨ e bounded self-adjoint operator acting in L2d x Ž0, `. and, 5 L 25 s p. Proof. It follows from Ž1.1. that L 2 is unitarily equivalent to A, which is a bounded self-adjoint operator. 2. THE SPECTRAL REPRESENTATION OF L 2 To find the spectral function of L 2 , we only need to find the spectral function of A. By using the Parseval relation where f, c g L2d x Žy`, `., we
20
BOUMENIR AND AL-SHUAIBI
have `
Ž Af , c . s H Af Ž x . c Ž x . dx y` `
s
Hy`k ) f Ž x . c Ž x . dx
s
Hy`k ) f Ž l. cˆ Ž l. d 2p
s
Hy`k ) f Ž l. cˆ Ž l. d 2p
s
Hy` fˆŽ l. cˆ Ž l. ˆk Ž l. d 2p .
`
$
l
`
$
l
l
`
Ž 2.1.
Let a be the increasing function defined by
ˆk Ž aŽ l . . s
p cosh p a Ž l .
s l,
a Ž l . - 0 for 0 - l - p ,
and so a is defined by
aŽ l . [
y1
p
ln
p l
q
(
p2 l2
y1 .
Ž 2.2.
We have from Ž2.1. p
Ž Af , c . s H l fˆŽ aŽ l . . ? cˆ Ž aŽ l . . d 0
aŽ l. 2p
0
Hp l fˆŽ yaŽ l. . cˆ Ž yaŽ l. . d
q s
yaŽ l . 2p
p
H0 l fˆŽ aŽ l. . ? cˆ Ž aŽ l. . 4 q fˆŽ yaŽ l . . ? cˆ Ž yaŽ l . . 4 d
aŽ l. 2p
.
PSEUDO-DIFFERENTIAL OPERATORS
21
The above Parseval equation can then be written in a matrix form as 1
p
Ž Af , c . sH l fˆŽ aŽ l . . , fˆŽ yaŽ l . . 0
2p
d
aŽ l. 0
0 aŽ l.
cˆ Ž a Ž l . . cˆ Ž yaŽ l . .
.
Ž 2.3. This only means that e i x aŽ l ., eyi x aŽ l.4 are the eigenfunctionals of A, the multiplicity of the spectrum is 2, and the associated spectral matrix is aŽ l . 0 2p 1
0 . aŽ l.
PROPOSITION 2.1. The spectral matrix of A is aŽ l . 0 2p 1
0 , aŽ l.
and the multiplicity of the spectrum s s supp daŽ l. s w0, p x is 2. Remark. The spectrum of A is continuous, and therefore the ‘‘eigenfunctions’’ are distributions, namely, e " i x aŽ l . that are outside L2d x Žy`, `., 5 e " i x aŽ l . 5 s `. Thus they are called eigenfunctionals or generalized functions Žsee w13, Vol. 3x.. Now we use our relation A s V L 2 Vy1 to derive the spectral representation for L 2 . We shall assume that f Ž x ., c Ž x . g C0 Ž0, `., where C0 Ž0, `. is the space of continuous functions with compact support, and so integrals are understood in the distributional sense, `
H0
L 2 f Ž x . c Ž x . dx s
p
H0 l f˜Ž l.
t
dL Ž l . f˜Ž l . ,
Ž 2.4.
where L is the spectral matrix associated with L 2 and f˜Ž l. is its transform. On the other hand, using Parseval equality, Ž2.3. leads to `
H0
L 2 f Ž x . c Ž x . dx s Ž AVf , Vc . s
p
H0 l Vfˆ Ž aŽ l. . , Vfˆ Ž yaŽ l. . =
aŽ l. 0
0 aŽ l.
1 2p
Vˆc Ž a Ž l . . Vˆc Ž yaŽ l . .
d
.
22
BOUMENIR AND AL-SHUAIBI
This simply means that the transform associated with L 2 is f˜Ž l . s
ˆ Ž aŽ l . . Vf
ž
ˆ Ž yaŽ l . . Vf
/
,
and the spectral matrix is given by L Ž l. s
aŽ l. 0 2p 1
0 . aŽ l.
Thus the spectrum of L 2 is continuous of multiplicity 2 and covers w0, p x. The corresponding eigenfunctionals are f˜Ž l . s
s
s
ž ž ž
ˆ Ž aŽ l. . Vf ˆ Ž yaŽ l . . Vf
/
H0` f Ž x . Vy1 e i x aŽ l. dx H0` f Ž x . Vy1 eyi x aŽ l . dx H0` f Ž x . xy
1 2
qi aŽ l .
dx
H0` f Ž x . xy
1 2
yi aŽ l .
dx
/
/
,
and thus the eigenfunctionals are xy1 r2qi aŽ l ., xy1r2yi aŽ l.4 . Let us denote the components of f˜Ž l. by `
Tq Ž f . Ž l . s
H0
Ty Ž f . Ž l . s
H0
`
f Ž x . xy1 r2qi aŽ l . dx f Ž x . xy1 r2yi aŽ l . dx.
Remark. Since f has a compact support, the integrals defining f˜ are well defined.
3. THE SPECTRAL RESOLUTION OF L It is readily seen that L is the square root of L 2 . For simplicity let us denote w Ž l . [ y 12 q iaŽ l2 . ,
l g Ž y'p , 'p . ,
PSEUDO-DIFFERENTIAL OPERATORS
23
and the classical Euler]Gamma function by G Ž x .. With the help of the eigenfunctionals of L 2 , we can reconstruct the eigenfunctionals of L . This is achieved with transmutations or what is called a transformation operator, a very useful device in inverse spectral problems Žsee w16x and w6x.. It is easy to see that the newly defined functions y Ž x, l . s G Ž 1 q w Ž l . . x wŽ l. q sign Ž l . < G Ž 1 q w Ž l . . < x wŽ l.
Ž 3.1.
satisfy L y s l y, where sign Ž l . s
½
q1 y1
l)0 l - 0,
and recall that l s signŽ l.< G Ž1 q w Ž l..<. Now use y to define the transform associated with L by ^
f Ž l. s
`
H0
f Ž x . y Ž x, l . dx
f g C0 Ž 0, ` . ,
and thus for l g Žy 'p , 'p ., ^
f Ž l . sG Ž 1qw Ž l . . ?Ty Ž f . Ž l2 . qsign Ž l . ? < G Ž 1qw Ž l . . < Tq Ž f . Ž l2 . .
Ž 3.2. It is clear that the multiplicity of the spectrum of L is either 1 or 2. We claim that Eq. Ž3.1. defines a complete system of eigenfunctionals, and so the multiplicity is only 1. Recall that the system of eigenfunctionals y Ž x, l. is complete if ^
f Ž l . s 0 for
l g Ž y'p , 'p . « f s 0
in L2d x Ž 0, ` . .
^
Using Ž3.2. for l ) 0 and l - 0, condition f Ž l. s 0 is equivalent to the following system: G Ž 1 q w . ? Ty Ž f . Ž l 2 . q < G Ž 1 q w . < ? Tq Ž f . Ž l 2 . s 0 G Ž 1 q w . ? Ty Ž f . Ž l2 . y < G Ž 1 q w . < ? Tq Ž f . Ž l2 . s 0. Since its determinant is nonzero, we have
Ž Ty Ž f . Ž l2 . , Tq Ž f . Ž l2 . . s f˜Ž l2 . s 0, that is, f s 0 in L2d x Ž0, `.. Hence the set y Ž x, l. for l g Žy 'p , 'p .4 is complete in L2d x Ž0, `..
24
BOUMENIR AND AL-SHUAIBI
We now deduce a spectral representation for L from Ž2.4.. The transition formula Ž3.2., together with the Parseval relation, implies for f, c g L2d x Ž0, `., since the domain of L 2 and L coincides with L2d x Ž0, `., that
Ž L f , Lc . s Ž L 2 f , c . . Upon using the Parseval equality associated with the operator L and L 2 , we obtain
Hy''pp l f Ž l. c Ž l. d r Ž l. ^
2
^
s
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
,
where r Ž l. is the spectral function associated with L , which is to be determined here. Recall that < G Ž1 q w .< 2 s l2 , and use Ž3.2. to obtain
Hy''pp l f Ž l. c Ž l. d r Ž l. 2
s
^
^
Hy''pp l G Ž 1 q w Ž l. . ? T 2
y
Ž f . Ž l2 .
qsign Ž l . ? < G Ž 1 q w Ž l . . < Tq Ž f . Ž l2 . 4 ? ? GŽ 1qwŽ l. . ?Ty Ž c. Ž l2. qsign Ž l. ?< G Ž 1qw Ž l. . < Tq Ž c. Ž l2. 4 d r Ž l . s
Hy''pp l T 4
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
Hy''pp signŽ l. ? G Ž w q 1. < G Ž 1 q w . < T
q
y
Ž f . Ž l 2 . Tq Ž c . Ž l 2 .
qG Ž 1 q w . < G Ž 1 q w . < Ty Ž f . Ž l2 . Tq Ž c . Ž l2 . 4 d r Ž l . . Since the second integrand is an odd function, we only need to assume that r Ž l. is an odd function for the above expression to reduce to
Hy''pp l f Ž l. c Ž l. d r Ž l. ^
2
s
^
Hy''pp l T 4
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
PSEUDO-DIFFERENTIAL OPERATORS
25
to obtain
Hy''pp l T 4
s
y
Ž f . Ž l 2 . T y Ž c . Ž l 2 . q Tq Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
.
Therefore we have
H0'p l t
2
4
s
y
Ž f . Ž l 2 . Ty Ž c . Ž l 2 . q T q Ž f . Ž l 2 . Tq Ž c . Ž l 2 . 4 d r Ž l .
p
H0 l T
q
Ž f . Ž l . Tq Ž c . Ž l . q Ty Ž f . Ž l . Ty Ž c . Ž l . d
aŽ l. 2p
.
Since the above relation holds for all f and c in C0 Ž0, `., which is dense in L2d x Ž0, `., the measures must be equal: 2 l2 d r Ž 'l . s s s s s s
l 2p
daŽ l .
0-l-p
l daŽ l . 2p
dl
dl
l 2pl'p 2 y l2
l
dl dl
2pl'p y l 2
1 2p'p 2 y l2
'l p'p 2 y l2
2
d'l
2'l d'l d'l ,
and hence
r 9 Ž 'l . s r 9 Ž l. s
d'l
'l 2pl2'p 2 y l2 1 2pl3'p 2 y l4
.
26
BOUMENIR AND AL-SHUAIBI
Thus we obtain
r 9 Ž l. s
1 2pl3'p 2 y l4
,
where 0 F l F 'p ,
and to recover r Ž l. over all of s , we need to recall that r Ž l. is an odd function,
r 9 Ž l. s
PROPOSITION 3.1. f Ž x. s
1 2p < l < 3'p 2 y l4
y'p F l F 'p .
Ž 3.3.
If f g C0 Ž0, `., then we ha¨ e 1
Hy''pp f Ž l. y Ž x, l. 2p < l< 'p ^
3
2
y l4
dl,
and F g H 2 l L2d x Ž0, `., L y1 F Ž x . s
1
Hy''pp F Ž l. y Ž x, l. 2p < l< l'p ^
3
2
y l4
dl.
Ž 3.4.
Recall that the spectral representations is not unique. A simple rearrangement will allow us to obtain an increasing spectral function. Indeed, we can define
m Ž l. [
½
rŽ l y p . rŽ l q p .
for l G 0 for l - 0.
Equation Ž3.4. opens the possibility of using regularization methods based on the spectral properties such as those found in w4x. It is clear that the spectral function has three points to regularize,
l s y'p , 0, and 'p .
4. ANALYTIC PSEUDO-DIFFERENTIAL OPERATORS The following theorem furnishes a real inversion formula for L y and unveils the nature of the unboundedness of the inverse Laplace transform operator. 1
PSEUDO-DIFFERENTIAL OPERATORS
THEOREM 4.1.
Let F g H 2 l L2d x Ž0, `.. Then
Ly F s 1
27
1
p
Vy1 cos Ž p D . V L F
where D [
d dx
.
Proof. It is known that convolution operators can be represented as differential operators. Let f g L2d x Žy`, `. be such that fˆ has a compact support; then the Fourier transform applied to Ž1.2. yields Af Ž x . ' k ) f Ž x . $
Af Ž l. s ˆ k Ž l . fˆ Ž l . p s fˆ Ž l . cosh pl
$
Ay1 f Ž l . s s
cosh pl
p
fˆ Ž l . $
1
cosh Ž p Ž yiD . . f Ž l . ,
p
where D s drdx. Thus by taking the inverse Fourier transform we have Ay1f Ž x . s
1
p
cos Ž p D . f Ž x . .
Ž 4.1.
Observe that since fˆ has a compact support, f is infinitely many times differentiable. Now that we have a representation for the operator Ay1 , we can deduce from Ž1.1. the inverse Ly2 s Vy1Ay1 V 1 s Vy1 cos Ž p D . V p Ly1 s
1
p
Ž 4.2.
Vy1 cos Ž p D . V L ,
which is valid if V L F is in the domain of cosŽp D .. We now show that F g H 2 l L2d x Ž0, `., i.e., in the range of Laplace transform L , is sufficient for V L F to be in the domain of cosŽp D .. Indeed, if F g H 2 l L2d x Ž0, `., then there exists f g L2d x Ž0, `. such that F s L f, and so VLF s VL 2f.
28
BOUMENIR AND AL-SHUAIBI
Upon using Ž1.1. we obtain V L F s AVf , which means that Vf is the domain of A, and V L F is in the range of A, and, in other words, V L F is in the domain of Ay1 . Hence Ž4.1. implies that V L F is in the domain of Ž1rp .cosŽp D ., i.e., F g H 2 l L2d x Ž 0, ` . «
1
p
cos Ž p D . V L F g L2d x Ž y`, ` . ,
and so Ž4.2. holds in H 2 l L2d x Ž0, `.. Equation Ž4.2. establishes a direct connection of the inverse Laplace transform with pseudo-differential operators, or differential operators of infinite order. This allows us to use many results related to approximation Žsee w11x.. It is easy to see that the ill-posedness of the inverse transform is really due to cosŽp D . only, since all of the remaining operators are bounded in Ž4.2., and hence any regularization should deal with cosŽp D . only. We recall that cosŽp D . is a bounded operator in the space ExpAŽ C . Žsee w10x and w11x., where Expr Ž C . s g Ž z . entire < g Ž z . < F M exp Ž r < z < . 4 . It is readily seen that Expr Ž C . is a Banach space with norm 5 u 5 r [ sup < u Ž z .
Recall that if r 1 -r 2 ; then we have the compact embedding Expr 1Ž C . ¨ Expr 2Ž C .. These compact embeddings help the construction of countably normed spaces, as an inductive limit, ExpA Ž C . s lim ind Expr Ž C . , r A
F Ž s.
6
Exp AŽ C .
6
and a sequence Fn is said to converge in ExpAŽ C ., i.e., FnŽ s . Expr Ž C . if there exists r - A, such that FnŽ s . F Ž s ., i.e., ;K compact sup < Fn Ž z . y F Ž z .
We now apply the above results to improve the inverse Laplace transform as defined by Ž4.2.. Recall that for an entire function, the Taylor series represents a translation, and so for a function t analytic in a
PSEUDO-DIFFERENTIAL OPERATORS
29
neighborhood of the point a,
t Ž a q h. s
hn
Ý nG0
n!
D nt Ž a .
s exp Ž hD . t Ž a . . Using the above representation we have cos Ž p D . g Ž x . s s
exp Ž ip D . q exp Ž yip D . 2 1 2
gŽ z.
Ž g Ž x q ip . q g Ž x y ip . . ,
Ž 4.3.
which is valid when g in defined on the real line and has an analytic extension in the strip V [
1 2
Ž V L F Ž x q ip . q V L F Ž x y ip . .
to hold. Thus we have THEOREM 4.2. v v
If
F g H l L2d x Ž0, `., V L F Ž x . has an analytic extension in the strip V [ < Im z < F p 4 , then 2
L y1 F Ž x . s Vy1
V L F Ž x q ip . q V L F Ž x y ip . 2p
.
Ž 4.4.
We now simplify formula Ž4.4. by using the Schwartz reflection principle Žsee w7x.. Indeed, observe that if F Ž s . is real for real s, then V L F Ž x . is also real for real x. Hence by the Schwartz reflection principle, applied in the strip V [
30
BOUMENIR AND AL-SHUAIBI
Hence we have proved COROLLARY 4.3. v v v
Let the following conditions hold:
F Ž s . is real for real s. F g H 2 l L2d x Ž0, `.. V L F Ž x . has an analytic extension in the strip V [ < Im z < F p 4 .
Then L y1 F Ž x . s
1
p
Vy1 Re V L F Ž x q ip . .
Ž 4.5.
Remark. In Ž4.4., the operators V and Vy1 do not cancel, as it would mean that cosŽp D . commutes with V.
5. EXAMPLES We now consider examples where F Ž s . is real for real s, and we shall use Ž4.5. to compute the inverse of the Laplace transform F. For simplicity we shall start with f g L2d x Ž0, `., to guarantee that F g H 2 l L2d x Ž0, `., i.e., it is in the range of the Laplace transform. EXAMPLE 1. Consider f Ž t . s Ž1r 't .w H Ž t y 1. y H Ž t y 36.x, which belongs to L2d x Ž0, `. and whose Laplace transform is F Ž s. s L f Ž s. s
'p 's
Er fc Ž 's . y Er fc Ž 6's . ,
where Er fcŽ's . is the error function Er fc Ž x . s 1 y
2
x
'p H0 exp Ž yt
2
. dt.
By construction, F is in the range of L , and so Corollary 4.3 can be applied to obtain f s L y1 F s Ž1rp .Vy1 Rew V L F Ž x q ip .x. Taking the Laplace again, we obtain L FŽ x. s
2
'x
arctan Ž 'x . y arctan
'x
ž / 6
,
PSEUDO-DIFFERENTIAL OPERATORS
31
which is an entire function of x, and so Theorem 4.2 applies: 1
Re V L F Ž x q ip . s
p
2
p
Re arctan y exp Ž x . y arctan
'
(
y1 36
exp Ž x . .
Using the previous equation, we have 1
p
1
Re V L F Ž x q ip . s
2
csign 1 y
ž
1 6
/
'expŽ x .
y csign 1 y exp Ž x . ,
ž
'
/
and so Vy1
1
p
Re V L F Ž t . s
1
csign 1 y
ž
2't
1
't
6
/
y csign Ž 1 y 't . . Ž 5.1.
Then, using the fact that
½
csign Ž t . s
if Re Ž t . G 0 if Re Ž t . - 0
1 y1
s 1 y 2 H Ž yt . , that is for t G 0, csign Ž 1 y 't . s 1 y 2 H Ž 't y 1 . s 1 y 2 H Ž t y 1. , from Ž5.1. we have Vy1
1
p
cos Ž p D . V L F Ž t . s s
1
2 H Ž t y 1 . y 2 H Ž tr36 y 1 .
2't 1
't
H Ž t y 1 . y H Ž t y 36 . .
We obtain the required result in the L2d x Ž0, `. sense Vy1
1
p
cos Ž p D . V L F Ž t . s f Ž t . .
EXAMPLE 2. Let f Ž t. s
½
0 y1rt
tF1 t ) 1,
32
BOUMENIR AND AL-SHUAIBI
which obviously belongs to the domain of L . Its Laplace transform is given by F Ž s . s L Ž s . s Ei Ž ys . , where Ei denotes the exponential integral defined by exp Ž t .
`
Ei Ž x . s PV
Hy`
t
dt.
Its Laplace transform is given by L F Ž s. s
y1
ln Ž 1 q s .
s
and V L F Ž x . s exp
yx
ž / 2
ln Ž 1 q exp Ž x . . .
Clearly, V L F Ž x . is entire in x. Then by the Schwartz reflection principle we have 1
p
cos Ž p D . V L F Ž x . s s
1
p 1
p
Re yI exp exp
yx
ž / 2
yx
ž / 2
ln Ž 1 y exp Ž x . .
arg Ž 1 y exp Ž x . . ,
which leads to Vy1
1
p
cos Ž p D . V L F Ž t . s
1
pt
arg Ž 1 y t . .
Recall that arg Ž j . s
½
0 p
if j ) 0 if j - 0.
Ž 5.2.
Thus we have recovered f in the L2d x Ž0, `. sense, Vy1
1
p
cos Ž p D . V L F Ž t . s
1 t
H Ž t y 1. ,
where H is the Heaviside step function. EXAMPLE 3. Let us take a simple function in L2d x Ž0, `., defined by a pulse f Ž t . s H Ž t y 1. y H Ž t y 2. .
PSEUDO-DIFFERENTIAL OPERATORS
33
Then F Ž s. s L f Ž s. 1
s
s
exp Ž ys . y exp Ž y2 s .
is entire and belongs to the range of L . Next we compute L F Ž x . s ln Ž x q 2 . y ln Ž x q 1 . , and so V L F Ž x . s exp
x
ž / 2
ln Ž 2 q exp Ž x . . y ln Ž 1 q exp Ž x . . .
By the Schwartz reflection principle we have 1
p
cos Ž p D . V L F Ž x . s s
1
p
y1
p
x
ln Ž 2 yexp Ž x . . y ln Ž 1 y exp Ž x . .
½ ž / ž / Ž
Re I exp exp
x
2
2
5
arg 2 y exp Ž x . . y arg Ž 1 y exp Ž x . . .
Thus Vy1
1
p
1
cos Ž p D . V L F Ž t . s
p
arg Ž 1 y t . y arg Ž 2 y t . ,
and upon using Ž5.2. we obtain in the L2d x Ž0, `. sense that Vy1
1
p
cos Ž p D . V L F Ž t . s H Ž t y 1 . y H Ž t y 2 . s f Ž t. .
The next example deals with the transform of the Sinc function. EXAMPLE 4. Let us denote f Ž t. s
sin Ž t . t
,
which obviously belongs to the domain of L . Then its Laplace is given by F Ž s. s
p 2
y arctan Ž s . .
34
BOUMENIR AND AL-SHUAIBI
The second iteration by Laplace Žsee w9x. gives L F Ž s. s
p 2s
y
sin Ž s . s
Ci Ž s . q cos Ž s .
Ssi Ž s . s
.
Transforming the result by V yields p VL FŽ x. s y sin Ž exp Ž x . . Ci Ž exp Ž x . . 2 qcos Ž exp Ž x . . Ssi Ž exp Ž x . . exp y
x
ž / 2
.
Ž 5.3.
We now recall that the sine integral functions Si Ž x . s
x
H0
sin Ž t .
Ssi Ž x . s Si Ž x . y
dt
t
p 2
are real functions for real x, and Ci Ž x . s g q ln Ž x . q
x
H0
cos Ž t . y 1 t
dt
is also real if x ) 0. Thus we deduce that V L F is an entire function in x that is real for real x, and thus the Schwartz reflection principle implies that 1
p
cos Ž p D . V L F Ž x . s
1
p
Re yI exp y
½
x
p
2
2
ž /
y sin Ž yexp Ž x . . Ci Ž yexp Ž x . .
qcos Ž yexp Ž x . . Ssi Ž yexp Ž x . . s
1
p
exp y
x
ž / 2
Im
p 2
q sin Ž exp Ž x . . Ci Ž yexp Ž x . . qcos Ž exp Ž x . . Ssi Ž yexp Ž x . .
s s
1
exp y
x
sin Ž exp Ž x . . Im Ci Ž yexp Ž x . .
ž / p ž / p ž / Ž 1
exp y
s exp y
x
2
2 x
2
sin Ž exp Ž x . . Im ln Ž yexp Ž x . .
sin exp Ž x . . .
5
PSEUDO-DIFFERENTIAL OPERATORS
35
Applying Vy1 leads us to Vy1
1
p
cos Ž p D . V L F Ž t . s
sin Ž t . t
,
which holds in the L2d x Ž0, `. sense. EXAMPLE 5. Consider f Ž t . s 3t 2 expŽyt .; then its Laplace transform is F Ž s. s
6
Ž s q 1.
3
,
and L F Ž s . s 3 y 3s q 3s 2 exp Ž s . Ei Ž 1, s . . Applying the V operator yields V L F Ž x . s 3 exp Ž x . 1 y exp Ž x . q Ei Ž 1, exp Ž x . . exp Ž exp Ž x . q 2 x .
'
and 1
p
cos Ž p D . V L F Ž x . s 3 exp
5
x exp Ž yexp Ž x . . ,
ž / 2
which implies in the L2d x Ž0, `. sense that Vy1
1
p
cos Ž p D . V L F Ž t . s 3t 2 exp Ž yt . .
6. CONCLUSION Using operational calculus and pseudo-differential operators, we have reduced the inverse Laplace transform to simple operations. Indeed, all we need to do is to take the Laplace transform L F Ž s . where F Ž s . is the given transform and then use translation operators with a change of variable. One should recall that similar expressions were obtained in w19x, except for a complicated symbol involving the inverse of the Euler]Gamma function, which remained a major obstacle for simple computational applications. This difficulty is removed in our case, since the symbol cosŽp D . is entire and can be expressed in terms of simple translation operators.
ACKNOWLEDGMENT The authors gratefully acknowledge the support provided by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.
36
BOUMENIR AND AL-SHUAIBI
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