A generalized Laplace transform of generalized functions

Analysis Mathematica, 18 (1992), 139--151

A generalized Laplace transform of generalized functions

A. K. MAHATO and K. M. SAKSENA

1. The integral transform

F(s) = 2 -~/~ / (st)~e-Sq~v(r f ( t ) dt 0

where N, denotes Weber's parabolic cylinder function, studied by TIVARI [4], has recently been extended by the authors to a class of generalized functions. This transform reduces to Laplace transform for 2 = v = 0 and will be called Weber transform. In this paper we have proved an inversion formula for the generalized Weber transform and a uniqueness theorem for it. A structure formula for a class of Weber transformable generalized functions has also been obtained.

Let u = x ; t e - " / '~ (1/2-~

where

~,(z) = e-:/iz"{1 n ( n - - 1 ) n ( n - 1 ) ( n - 2 ) ( n - - 3 ) } 2z ~ + 2 . 4 z 4 . . . .

Using a differential relation for Weber's parabolic cylinder function, we have

since

x(X/,.)+;t d r 12 d ] 1 dx Ix I .-~x {X-~U} = .~ xae-~/,N,+,(1/2- ~

d n

dz" [e-:/4Nv(z)] = ( - 1)"e-:/4Nv+"(z)' n = 1, 2 . . . . .

(See [2, p. 119].)

Further, if we define an operator ~,/~ by

Aa, xq~(x) = xt/S+a[D~xl/SD~{x-~qT(x)}]

Received October 15, 1990.

140 A.K. Mahato and K. M. Saksena

d where Dx---~--x ' then it has been shown in [2] that

s (st)~e_~,12N,+~(]/~--~) ' Aa,, {(st) a e-,,/~N, (]/~-i)} = ~-

.4"~, , { (s t )~e-~ ' l~ ,O/~-~- i )} = -ff ( s t ) a e - ~ ' / ~ . ~ . ( W s - - ~ , n = O, 1 . . . . .

and that

Ana,, {(st)a e-S'/ '~, (]/ 2s-"~} < ~

for large and small t provided that Re s>0 . Let e and /? be real numbers and 2 a complex number with Re 2>0 . Let

K,,p(I) be the set of all those complex valued smooth functions q~(t) defined on I(O, ~) for which the functionals

6~,tj,,(tp)= sup le"tP+"A~aq)(t)l

are finite for n=0 , 1 . . . . . With the usual pointwise operations of addition of functions and multiplica-

tion by a complex number, K,,~(I) is a linear space. The collection ~r {6,,p.,},= 0 x also forms a countable multinorm on K,,p(I) and equipped with the topology generated by this multinorm, K,,p (I) becomes a countably multinormed space which is also complete. It also satisfies all conditions for being a testing function space and its dual K',p (I) is also complete. We call f a Weber-transformable generalized function if it is a member of K'~,p(I). The Weber transform F(s) of fEK'p( I ) is defined by

F(s) = (~a, , f )(s) = (f(t) , co(st)) where

co(st) = 2-vl2(st)*e-*'/2~v(l/2s-~, Res > e, R e 2 + / ? > 0 and sCfay.

The region ~2f is defined by

t? s = s : R e s z ~ s , s r

where a s is a real number (possibly a ; = - ~ ) such that fEK't j (I ) for every a>ajr and fCK'~,r for e < a f .

D(I) will denote the standard countable union space (see [6, pp. 32, 33]) of the countably multinormed spaces Dk(I) of all complex-valued smooth functions defined on I(O, ~o) which vanish on those points of I which are not in a compact subset K o f / , with seminorms defined dy

~k(q~) = sup ID~o(OI, (PEDk(1), t E I

Generalized Laplace transform of generalized functions 141

and the topology generated by the countable multinorm {~k}kr assigned to the corresponding linear space with usual pointwise operations of addition and multiplication of functions.

2. We now prove an inversion formula which determines the restriction to D(I) of any ~a,v-transformable generalized function from its Weber transform and then give a weak version of a uniqueness theorem.

Let us first prove a few lemmas.

L e m m a 2.1. I f fEK'.p(I), then

o 0 where

o~(xu) = 2-'lg(xu)Xe-XUl=~v(l/2x-~, R e s > 0t, R e a + f l > 0.

P r o o f . It is clear that

x - ' ( f (u), o (xu)) = ( f (u),

Let 1o, ~ (x) denote both the function

x_ 0 /0,=(x) = if 0 < x <=,,

and the corresponding generalized function belonging to K'=,p(1). Then using the definition of product of generalized.functions [6, p 121], we have

(2.1) (lo.~(x)f(u), x-'o~(xu)) = (lo,~(x), ( f(u), x-~oa(xu))} =

(2.2) = f (f(u), x- 'o~(xu)) dx, o

Step (2.1) is justified since x-'o~(xu)EK,.B(I ). Step (2.2) is obvious since lo.=(x) is a regular generalized function.

Since the product of generalized functions is commutative, the left-hand side of (2.1) can be written in the form

(2.3) ( f (u) lo, = (x), x-'o~(xu) ) =

(2.4) = ( f (u), (lo, o o (x), x-'o~(xu))) =

o

Thus (2.2)=(2.5) which proves the lemma.


L e m m a 2.2. Let ~pE D(I) and r be a fixed real number. Let

~(s) = [ t-Scp(t)dt 0

where s=c+iT, c fixed, and max(as, 1 ) < c < ~ o , - ~ o < T < ~ . I f fEK'p(I) and 0e<0, then

(2 6) 2"-'~] -,f (f(u)'us-t>~b(s)dT= f(u), ~ - , f ~,-,~,(s)a .

P r o o f . For ~p(t)=0, the proof is trivial. Let ~p(t)~0 and let

(2.7) (f(u), u "-a) = A(s).

(2.7) is justified since u*-aEK~.p(/) for R e s > l - f l . It can be seen that A(s) is analytic for all s for which max (a S, 1 )<Res< oo and ~(s) is also analytic for all finite values of s.

Thus the left-hand side of (2.6) is an integral with an integrand analytic over a finite region and hence converges uniformly. Now

]e"'ua+" A~,,, {~---.~ _f, u'-'r (s) aT}l = [e"u#+" ~ _,f' {Aa..~-~} ~,(s)dT[=

i 1 ( = e"ua+"~-~n . . .

- - r

- ~ 'n - , l e " u P + S - t l ( - A + s - I) - 2 + s - 3 . . .

As " 3 2n+ 1

- - r

is finite and

we see that

sup le~"uP+~-I I <oo for R e s > l - p , ~ < O, O - c u ~ : o o

sup e~uP+'A'~. "u~-~(s)dT I _<.; }<-


! proving that f u~-~$(s)dT as a function of u belongs to K,.p(I). Hence the right-

hand side of (2.6) is also meaningful. Now to prove the equality, let us partition the path of integration on the straight

line from C- i r to C +ir into m subintervals each of length 2r/m. Let sp = C + iTp be a point in the p-th interval. We can write

1 P

f (f(u), ~-,)r = (2.8) 2-K

= h m Z x'--(f(u), u'~-l)r 2r m~ p=l Zn m

= lim u), us,, g/(s n . t a b o o

Let us set 2r

p=l

I f we can show that the sum within the last expression in (2.8) converges in K~,p(I)

to f ~-xO(s)dT, then equality (2.6) will be proved.

Let us consider B(u, m) where

(2.9) B(u, m) = :"u#+"&. [v.,(u)- f u'-x~(s) dr] =

=e"u'+"[A'a,.lVm(u)-A'a.,, f u*-Xg/(s)dT}] =

3 ~ p=l 2

"u"-'-"d/(s')2rl--e"uP+" m , / (--~L+s-- 1 ) ( - -2+s- - 23I) ''" ~ r

. . . ( - 2 + s - n ) - 2 + s - 2

We have to show that B(u, m) converges uniformly to zero on 0 < u < ~ as m ~ . For e<0 , we see that

tends uniformly to zero on -r~_T~_r as u - - ~ . Consequently, given 8>0, there

144 A . K . Mahato and K. M. Saksena

exists u ' > 0 such that for u > u ' > 0 and - r = T = r we have

l e ~ " u P + ~ - z ( - 2 + s - - 1 ) ( - 2 + s - - 3 ) ' " ( ' 2 + s - n ) ( - ' ~ + s " 2n+l }

- - r

as /[~(~)dTI is finite and # 0 due to q~(t)#0. It follows that - - r

1 u , _ ~ O ( s ) d r < 8 (2.10) sup e~"ua+"A~,u ~ -~. U : ~ U I

Also for all m, ' 2 r

(2.11) sup le'"ua+"AL. {Vm(u)}l < 3[ f IO(s)arl]-l ---~ ~ 1O(so)l. I t z ~ U t ~ r =

2e Thus, there exists m0 such that for m >m0, the right-hand side is bounded by -~--.

By (2.10) and (2.11), we see that IB(u, m)l<* for m>mo, u>u'. Let us now consider the range 0 < u < u' with a fixed s in max (cry, 1) < Re s < oo,

We see that

3 ( - 2 + s - 2 -) O(s) u a + , _ l ( _ 2 + s _ l ) ( _ 2 + s _ . ~ ) . . . ( _ 2 + s _ n ) 2n +._.....~1

is a uniformly continuous function of (u, T) for 0 < u < u ' and -r<=T<r. This together with (2.9) shows that there exists mz such that for all m>m~ we have IB(u,m)l<8 on 0<u" as well.

Thus, when m>max(m0, ml), we have IB(u, m)l<e uniformly on 0<u<~o . Hence Lemma 2.2 is proved.

L e m m a 2.3. Let (i) q~D(I), (ii) ~, fl, c and r be real numbers such that max (cry, 1 ) < c < co and cr Then

~o ~ sin r log u l ! q~(t) ( t ) U-- dt--qg(u)

ulog-- T

in K~, ~ (I) as r ~ oo.

P r o o f . Let

1 ~ ( t ) c s i n r l ~ I = -~- /~o( t ) u

ulog T

dt.


Putting u = t e ~ in I we have

Hence

since

Let

I=lf _ _ ~o(ue_~) e(C_l) ~ sin rx dx. I[ X

, f I - ~p(u) = -~ [et~-x)Xg(ue -x) -- cp(u)] sin rx dx - - r X

f sin rx dx = ~. X

- - o o

1 f [e(C_l)xtp(ue_X)_~o(u) ] sinrXdx" O , ( u ) = e ~ u P + " a " ~ , . - s _ ~ x

Our lemma will be proved if we prove that O,(u)-~O as r ~ co uniformly on 0 < u < ~ o ; n=0 , 1, . . . . Taking the differential operator inside the integral sign,

we have

O,(u) 1 e, Uup+ . f [e<C_l,xn. ~ . tp(ue_X)_Al , .~o(u) ] sin rX dx

I + f + =Ii+I,+I3, say.

Let us consider/~ first and set

R(x , u) = e ~" u p+" 1 [e(~_~)XA~, u ~o(ue -~) - A~,. e (x)l. X

By virtue of our supposition, it is evident that R(x , u) is a continuous function of (x, u) for all u in O < u < ~ and x r Also, by 1' Hospital's rule,

lim R(x, u) = lim e~"uP+"D~[e(~-t)XA~,.q~(ue-~)]. x ~ O x ~ O

Hence assigning the value

e'U uP+nDx [e(~-1)~A~, u ~o (ue-X)][x= o

to R(0, u), we see that R(x , u) is a continuous function o f (x, u) in - 6 < x < 6 ; 0 < u < oo and since r is smooth, R(x , u) is bounded, say, by K. Hence, for any e>0 , there exists 6 > 0 so small that

,12(u), = 1 / R ( x , u ) s i n r x d x [ =

I-> i I , -~ IR(x,u)sinrxldx ~_-~_~


or,

if 6 is fixed such that 6 < 7re/K. Now let us consider Iz(u).

I , ( u ) < = x

Ii(u) = le* 'ua+" f e(C-X)XA"a .9(ue -~) sin rx d x - ' X

- - o o

We have

1 e~tU ga+n ) ~ - - - A"~,uq~(u) s inrx dx = Jz(u)-J~(u), say. ff x

- - o o

1 -fa sin z dz. J~(u) = ~ e"ut~+"[A"x,.q~(u)] z ~ o o

sinz Now, as e"ut~+"A~.,,q~(u) is bounded on 0 < u < . o and [

--oo# Z

d2(u) tends uniformly to zero in 0 < u < o o as r--.oo, since

- '* sin z lim f ~ d z = O.

By integrating by parts,

dz is convergent,

Jl(u) = !e'.uP+"n [ eC*-"x _ _ A~,.cp(ue-~) c~ -6 + r J - o .

1 ~ - . ) 6 { e (c-~)~ } + - - e uS+ cosrxDx A~,.~o(ue -~) dx.

~r x - - o o

Since cp(u)ED(I), i.e., it is of compact support, and c > l , we may write

(2.12) fl (u) = _~llrr e~u up +" + e-(c-1)aA~, ~ ~o (ue a) cos rff +

-~ [ et~-x)~ ] + l-~-e"ua+" f cosrxDx A~,.tp(ue -x) dx.

l i t x

The first term in (2.12) tends uniformly to zero in 0 < u < ~ as r - ~ since ff and


~ U n - - X " c are fixed and e Aa.u~O(ue ) is a bounded function of u in O < u < ~. Also,

e(c-1)x ] e ~ uP+nDx A~,. q~(ue-O = x

1 = e'Uu ~+" - ~ ( c x - - x - - 1)etr

+e~Uup+n e (c-1)x 0 x Ox [A~.,~(ue-X)].

Since each term is a bounded function of u and x in 0 < u < 0% - o o < x < - 6 , the second term in the right-hand side of (2.12) also tends uniformly to zero as r---o~.

Thus, we see that Jt(u)-~O as r-~ o~ and combined with the fact that J~(u)-,O as r--,oo, we see that Ii(u)---0 uniformly in 0 < u < o o as r--oo.

Similarly, we can prove that Ia(u)-~O uniformly in 0 < u < ~ as r-~oo. Collecting all these results, we conclude that O,(u)-~O as r-~oo, uniformly

in 0 < u < o~. Hence Lemma 2.3 is proved.

3. Complex inversion formula

T h e o r e m . Let

(i) f~K~,a(I ), (ii) F(x) = (f(u),o~(xu)), andlet (iii) a, fl, c berealnumbers, such that

max(os , 1 ) < c < ~ o , a < 0 , R e ( 2 , s + l ) > 0 ,

R e ( 2 - s - 2 + 3 ) > 0 , s = c + i Z R e s > u , and R e A + f l > 0 .

Then,for ~p(y)6 D(I),

where ~(s)= f x-SF(x)dx. o

as r --. oo.

148 . A.K. Mahato and K. M. Saksena

P r o o f . Setting

.a'(s) =

3 v =) F(~.4 2 2

FQ, + 1 - s)F (2 + 3 - s ) '

the theorem will be proved by justifying the following steps:

c + i r

(3.1) ~ f c - - ir

dll(s) t -= }P(s) ds, r (t)) =

(3.2) - 1 c + i r

= / "2-~7c_f "~r176 dt=

<., : • i ,: 2re JI(s)~(s t-stp(t)dtdT= (s = c +iT) - - r 0

r c o

1 f.a(=){f x- 'F (=)~=} f t-=~(,)atdr= (3.4) - 2= - r 0 0

(3.5) 1

1

i ,= (,> { [ : . <: <~ =<xu)> ==) f ,-. <o, ) <,~: - - r 0 0

(3.6) f ~a(=)(ffu), f :=<o(xu)a=) f t-'Io(t)dtdT= --I 0 0

(3.7)

(3.8)

- - r 0

1 " "0 dT) i ( u ) . ~ j . u . - . I ,-.<o(,)~, =

( i " ) 1 f u=_X (3.9) = f(u),-~n q~(t) t-=dTdt = --F

(3.10) ( f (u) '2~/q~( t ) ( t} :s inr lOgt 1 U d ~

u log T

(3.11) = (f(u), ep(u)).

Since the integral in (3.1) is a continuous function of t, and q~(t) is a smooth function of compact support in (0, co), (3.1) implies (3.2). As the imegrand in (3.2) is continuous on a closed and bounded domain of integration, we can change the


order of integration in (3.2) to obtain (3.3). Furthermore, obvious and (3.6) is justified by Lemma 2.1.

Now we have

? 2 - ~/~ x - s (xu)a e - x.l~ ~ , ( 2 ~ ) dx = 0

(3.4) and (3.5) are

= US --1 f (X/./)(A + 514- s)-- 3/2 e - x,,/~ W.q2 + 1/4, - 1/4 (xu) d(xu) =, 0

by WHITTAKER and WATSON [5, p. 347],

provided

F O , - s + l ) F O . - - s + 3 ) ~_ U s - 1

( v 3) Re (2 -- s + l) > 0, Re 2 - s - ~ - + ~ - > 0

(see [1, p. 337]). Hence (3.7) is a simplification of (3.6). (3.8) is justified by Lemma 2.2. As the integral in (3.8) converges uniformly, we can change the order of integration to obtain (3.9). After simplification, (3.9) reduces to (3.10). Lemma 2.3 shows that the integral (3.10) converges in K,,p(1) to tp(u) as r-~ co) uniformly in 0 < u < co and (3.10) implies (3.11). So, the theorem is proved.

' I 4. Uniqueness theorem. Let f , gEK~.p( ) and

(i) F(s) = ( ~ , , f ) ( s ) , f6t2y;

(ii) G(s)=(~a,~g)(s) , s6g2g; and

(iii) F ( s ) = G ( s ) for sEf2fNf2a.

Then f = g , in the sense o f equality in D'(I).

The above weak version of uniqueness is an immediate consequence of the inversion theorem.


5. Structure formula

Now we will give a structure formula for the restriction of an element fEK'~,a (1).

T h e o r e m 5.1. Let f be an arbitrary element of K" p ( I). Then there exist bounded measurable functions g,(x) (x>0) , r=0 , 1 . . . . . 2q+ 1, q being a nonnegative integer depending on f, such that for an arbitrary q~ED(I) we have

2 q + l t

(5.1) ( f ' <P)= ~,--~o (-1) 'D( '+x) / " g'(t)e'*tPQ'(t)dt' q)(t))

where D indicates distributional derivative, Q,(t) is a polynomial in t and a is a positive number.

P r o o f . In view of the boundedness property of generalized functions, there exist a positive constant C and a nonnegative integer q such that for all tpED(I),

I(f, ~0)l <-- C max ~.p,k(~0) <= C max sup le"tP+"A~ ,~o(t)l, O~--ke~q O ~ k ~ _ q O ~ t ~ o o '

o r

~k

[~f, (o)1 -~ C max sup le"t p Z P,(t)cP(')(t)[ =< O~_k~_q O < t - < oo r=O

2k

_-< C max sup f D,Z [e"tP O~--k~--q O~--t~--~ t

oo 2 k §

= < C m a x sup f Z le~ttaQ,(t)q~(')(t)l dt,

where P~(t) and Q,(t) are some polynomials in t of degree r. Hence,

(5.2) (f , q)) ~_ C ~ x ;[e~t t# Q,(t)tpf,)(t)ldt. r : O 0

Consequently, in view of the Riesz representation theorem and the Hahn--Banach theorem, there exist bounded measurable functions g,(x), r=O, 1 . . . . . 2q+ 1, defined over I(0, ~), satisfying

2q-i- l

(f , q~) = Z (g,(t),e~ttPQ,(t)tP(')(t)), r=O

(5.3)

o r

(5.4) 2q+1 t

Z (-1),(pc,+,, f g,(Oe"tPQ,(Odt, o(O). r~O a

Here the differentiation sign indicates differentiation in the distributional sense and a is some positive number. Expression (5.4) is obtained by integrating (5.3) by


parts, since it can easily be shown that the function

t

f g, (t) e ~t t p Q, (t) d t a

corresponds to a regular distribution in D'(1).

A c k n o w l e d g m e n t s . The authors are thankful to the referees for their helpful comments. The second author is also thankful to the University Grants Commission, New Delhi, for financial support.

References

[ 1] A. ERD~LYI, F. OBERHETTINGER and F. G. TmCOMI, Tables of integral transforms, Vol. I, McGraw Hill, 1953.

[2] A. ERD~LYI, F. OBERHETTINGER and F. G. Tmcoi I , Higher transcendental functions, Vol. II, McGraw Hill, 1953.

[3] A. K. MAHATO and K. M. SAKSENA, Some results for a generalized integral transform, Rend. Mat. Appl. (Rome) 11 (1991), 761--775.

[4] B. M. L. TIVARI, Study of certain generalized integral transforms involving Weber's parabolic cylinder function, Ph.D. thesis, Agra University, 1966.

[5] E. T. WHITTAKER and G. N. WATSON, .4 course of modern analysis, Cambridge University Press, 1963.

[6] A. H. ZEMANIAN, Generalized integral transformations, Interscience, 1968.

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5 Analysis Mathemafica

A generalized Laplace transform of generalized functions

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