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On Kµ Transformation of DistributionsJ.A. Barrios a & J.J. Betancor aa Dep. de Economia Aplicada , Univ. de La Laguna , Tenerife, Spainb Dep. de Análisis Matemático , Univ. de La Laguna , Tenerife, SpainPublished online: 10 May 2007.

To cite this article: J.A. Barrios & J.J. Betancor (1991) On Kµ Transformation of Distributions, Applicable Analysis: AnInternational Journal, 42:1-4, 175-197, DOI: 10.1080/00036819108840040

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0 1991 Gordon and Breach Scimcc Publishers S.A. Printed in the United Kingdom

On Kp Transformation of Distributions

Communicated by R. P. Gilbert

J.A. BARRIOS* AND J.J. BETANCOR"

* Dep. de Economia Aplicada, Univ. de La Laguna, Tenerife, Spain.

0 Dep. de Anllisis Matemltico, Univ. de La Laguna, Tenerife, Spain.

Abs t rac t In this paper the integral K- transformation, also known as Cc

Mei jer transformation, is defined on new spaces of generalized functions. We use a procedure based on adjoint operators. The Mellin integral transformation plays an important role in our study and i t allows t o show that the classical K transformation is an isomorphism

P between certain Frkchet function spaces introduced herein. We establish connections between our generalized , K transformation and the

Cc well-known K transform studied by A. H. Zemanian.

Cc KEY WORDS: Integral K, transformation, Bessel function, Mellin transformation, generalized functions (Received for Publication 5 March 1990)

1. INTRODUCTION .

A generalization of the Laplace transformation defined by

where K (2) denotes the modified Bessel function of third kind and order p Cc

([201), was introduced by C.S. Meijer [lo]. The integral transformation (1)

is usually called Meijer transformation or also K -transformation. The last P

author established an inversion formula fo r the K transformation, when P

~~1s:. Later A.H. Zemanian [201 extended the above inversion formula t o

values of order p>'. 2

175

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176 I. A. BARRIOS AND J. J. BETANCOR

In the last years many authors have studied the Meijer transformation

and several generalizations of this one (see, for example, R.P. Boas 111,

121, A. Erdelyi [61, A.H. Zemanian 1201, C. Nasim [I21 and K. Soni and R.P.

Soni [161).

Apparently, A.H. Zemanian [19] was the f i r s t to study the K Cc

transformation on certain spaces of distributions. For every a ts lR and phO,

he introduced Frkchet spaces denoted by X The dual space of X is ~ c , a' ~c , a

represented as usual by X' The generalized K transform K' f of f E X' L a ' Cc Cc Cc, a

is defined a s follows

where hZf={sclC : sg(-co,OJ and Re s>c ). The number c is less or equal than a f f

and is called the abscissa of definition of K'f and nf is the region of Cc

definition of the K -transform of f . In [I91 A.H. Zemanian also proved P

inversion formulas fo r the generalized K transformation. Cc

More recently A. Schuitman [I41 has investigated a variant of the K Cc

transformation over several locally convex spaces of functions. He

considered said integral transformation a s a special case of convolution

transform.

This paper is inspired by ideas of A. Schuitman (1131 and 1141) and our

purpose is t o define the K -transformation on new spaces of distributions. Cc

We shall employ an approach essentially different to the one used by A.H.

Zemanian [191 and (2) will be a special case of the definition adopted

herein. This paper is organized as follows. We shall introduce in Section 2

new FrCchet spaces of functions and we shall investigate the behaviour of

the Mellin integral transformation on them. Section 3 will be devoted t o

study a mapping denoted by T that will be needed to analyze the P

K -transformation on the spaces introduced in Section 2. This we shall Cc

develop in Section 4. In Section 5 we shall define the K transformation on P

certain spaces of distributions. Finally, a distributional differential

equation involving the operator S will be solved in Section 6 by using the Cc

generalized K transformation. Cc

Some properties of the function K that will be useful in the sequel Cc

a re recalled now.

The behaviours of K near the origin and the infinity are the following fJ

ones (see [201, p. 173). For C( e lR and c > 0,

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ON K, TRANSFORMATION OF DISTRIBUTIONS 177

uniformly for arg z in the indicated interval. Henceforth arg z denotes the

principal argument of z and arg ze[-n,n). Moreover, for P O ,

By defining as usual the Mellin integral transform Mf of f by

then (see [81)

1 for Re s<p- and pkO.

2

2. SOME FUNCTION SPACES AND THE MELLIN TRANSFORMATION.

In this section we introduce new Frkchet spaces of functions and we study

the behaviour of the Mellin integral transformation on them. Throughout this

section 6 E [O,K). We define the sets of complex numbers G6={taC: larg tlSt91

and ~ i = { t € C : 1 arg t 1 <6} . Note that 0 1 G6. 6 6

We firstly define the definition of the spaces T (-m,m) and S (-m,m)

introduced by A. Scuitman I141. 6 A function +(t) is in T (-m,m) if, and only if,

dk i ) )(t) is a holomorphic function over G; and +(t) can be d t . .

continuously extended from G: to G6, for LEN, provided that 6>0; )(t) is an

infinitely differentiable function for t E (0,m) when 6=0.

ii) for every meN and k d

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J. A. BARRIOS AND J. J. BETANCOR

k d m ()I = sup (3 )(t) 1 < m.

We can see that the topology generated by the family of seminorms

} "p k meN, LEI is the same that the one induced by the system of seminorms

6 "n}naN where < was defined in 1141 p.9. As i t i s well-known T (-m,m) is a

6 6 Frechet space. The dual space of T (-m,m) is denoted by (T (-m,m))' and is

endowed with the weak topology. 6

The following properties of the space T (-m,m) will be useful in the

sequel.

Proposition 1 : If f ( t ) is a rational function that has not poles into 6

G6 then the mapping ) I) is continuous from T (-a,.) into itself.

PROOF : For every kaZ and mcN, we get

Moreover, there exist rl and ni belong to Z and Mi and Ni>O, for every

i-0,. . . ,m such that

n i d m-i

It I%) f( t )1 s N l , for l t l z l a n d t E G6.

Hence,

k-nl k-ri i sup 1 tkdm(f(t))(t)l 1 a M (sup 1 t 6 )(t) 1 + sup 1 t 6 ((t) 1 ) taG6 i =o teG6 teG6

Thus the proof is finished.

Proposition 2 : Let f (x ) be a locally integrable function on (0,m). If 1

there exist n, r E 2 such that io 1 f ( t ) 1 tndt < m and C l f ( t ) 1 trdt < m, then f

6 defines a regular distribution in (T (-m,m))' through

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ON K, TRANSFORMATION OF DISTRIBUTIONS

PROOF : I t i s sufficient t o see that

6 f o r every # E T (-m,m). H 6 S (-mlm) i s defined a s the space of entire functions @(s) f o r which

r (a) = s u p [sk e61sl@(s)l < m , for every m,k E IN mlk s ~ V ( m )

where V(m) denotes the s t r ip {SEC : I Re s Ism). 6 The topology of S (-m,m) is generated by the collection of seminorms

6 {=m, k'm, ~ E H '

Thus S (-m,m) i s Fr6chet space. Notice that in (6) exp(6l s 1 ) can be replaced by e '1 Imsl and the resulting family of norms i s equivalent

to {=m1 k'm, keIN' The following theorem i s a special case of Theorem 7 in 1141 and i t

6 6 establishes an important relation between the spaces T (-m,m) and S (-m,m).

Theorem 1 : The Mellin integral transformation is an isomorphism from 6 6 T (-m,m) onto S (-m,m)

Let now {an):=l be a monotone decreasing sequence of real numbers such

tha t inf{ak-ak+l:k=l,Z,. . . )>0. We introduce the space qB({a )m that n n=l

consists of all complex valued functions $(t) fulfilling

dk i ) @(t) i s a holomorphic function over G: and +It) can be d t

continuously extended from ti t o Gel f o r kW,provided tha t 6>O; $(t) is an

infinitely differentiable function fo r t E (0,m) when 6=0.

ii) f o r every m,k E N

k a ($1 = sup I tm 1 (t l+aj+l-a.i 8 ~ ) (t I $ ( ~

tEG6 1.1 d t

k

Here must be understood a s 1 when k=O. J -1

The sequence { ; Y ~ , k}m, kEIN is a separating collection

< m

of seminorms which

generates the topolog~ of Sb({an}b1). Thus S6({an}:=l) i s a Frkchet space.

The dual space of 5*({an}bl) i s denoted by (S6({a n n=1 )I1 and i t i s

equipped with the weak topology.

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180 J. A. BARRIOS AND J. J. BETANCOR

6 A useful property of 5 ({an}:=1) that can be shown as Proposition 2 is

the following one

Proposition 3 : Let f ( t ) be a locally integrable function on (0,m). If -a -n-al

[ l I f ( t ) l t ld t < m and [ l f ( t ) l t dt < m, fo r some n E M, then f

6 generates a regular distribution in (9 by

<f(t),*(t)> = c ( t ) $ ( t ) d t , for $ E 5@({a n n=1 1.

6 m Let c>O such that c<inf {ak-ak+l: k=l,2,. . . }. Now define Uc({an}n=l) t o

be the collection of complex functions @(s) satisfying

i) @(s) is a meromorphic function on C that can have a t most simple

poles in s=a n=1,2, ... . n ' ii) for every m,k E N

k

where W~(m,k)=~s~C:m+~+ak+l~Resrm+l+~+ak+l~. Here n must be again J =l

understood a s 1 when k=O.

We supply ~z ({a~} :=~) with the topology generated by the family of

seminorms {w ' } 6

m, k m, kcN' It inmediately follows that ~&l{a~}:=~) is a Frechet

space. Moreover the space ~ : ( { a } ~ - ) is not depending on 8 E n n-1 6 m

(0,inf{ak-ak+l:k=1,2 ,...)I. Hence in the sequel we will denote U ({a } 1, 6

n n=1 w and W(m,k) instead of ~&({a~}:=~) , w:,~ and Wc(m,kl.

m,k The Mellin integral transformation allows again t o establish a

connection between the spaces h{an}:=l) and U6({a n lm- n-1 1. Theorem 2 : Let M be the mapping defined by

where (M$l(s) is the meromorphic function on Q: that has at most simple poles

in s=a n=1,2, ... and such that (MJl)(s)=Sn{$)(s), for Re s>al. M is an n '

6 isomorphism from 5 ({a 1 onto %*({a }" 1.

n n=l n n=l PROOF : Assume that 6 ~ ( O , n ) , if 6=0 the proof is similar.

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ON K, TRANSFORMATION OF DISTRIBUTIONS 181

CO

Let $ be in !)'({a n }* n=1 1 and define P1(s) = Ill{$l(s) = J ts?,h(t)dt. 0

We f i rs t ly note tha t al(s) is holomorphic fo r Re s>al.

Moreover, integrating by par ts we get

a = -- I &(t I*( t ) )d t , f o r ~e s>al s-a l o

1 d a By denoting Q2(s) = cs-a -(t d t l*(t))dt, Q2(s) i s a holomorphic

function fo r Re s>a2, and (7) can be rewritten Pl(s) = -&-@2(s), provided 1

that Re s>al.

For every k=0,1,2,.. . , we define

P (s) is a holomorphic function for Re s>\+~ and the partial integrating k+l

rule leads t o

1 ak(s) = - s-ak Qk+l(s) , fo r Re s>ak

and

( -1 I rn+l cBk(s) = k + m

'k+m+l (s) , fo r Re s>ak and meH.

Therefore the function defined by

P1 ( s ) , fo r Re s>al

P(s) = 1 . ( -1 I m @m+l (s) , fo r Re s E (am+l,aml , s+a and meN m

n ( s - a . ) j = 1 J

is meromorphic in C and P(s) has a t most simple poles in s=an, n=1,2, ... . Moreover Q(s)=!?Jl{+Ns), f o r Re s>al. Hence (M$)(s)=@(s).

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182 J. A. BARRIOS AND J. J. BETANCOR

By virtue of (8) one has

for We ~ > a ~ + ~ and ka#.

By proceeding as in the proof of Theorem 1, we can obtain

for Re s>ak+l and keN, provided that 1 el IS.

Therefore for s~W(m,k) with m, k E M,

w [Re s-a k+l-l

k -a a sJ R I TP j 1) 1 ,t=Rexpcie2 d~ exp(*l s 1 d l l m s)

J -1 d t

0

when \ell S.

By choosing now suitable el, we get

k 1 k l+a -a a m+&-1 I e* I I ~ ( s ) , (s-a,) I .; J R d~ sup I , (t J+' j $)(t l+(t)) 1 +

J=I o taGg j = ~

k a + p e r c l ~ sup itr n ( t l+aj+i+l-aj i ) ( t l*(t)) 1, for s a I R ( ~ , L I

teG* J =I d t

with raN, such that m+c-r<-1.

Hence, there exists a positive constant M not depending on g such that

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ON K, TRANSFORMATION OF DISTRIBUTIONS

6 Then M is a continuous mapping from 4 ({an}:=,) into U6(an).

Let now @ E U6(an). We define with

c>al. As i t can be easily seen the integral defining $(t) i s not depending

on c, provided tha t c>al. Moreover $(t) is a holomorphic function on G: such

dK that --i;Jj(t) can be continuously extended t o G*, fo r each keK d t On the other hand, by differentiating under the integral sign we obtain

k a tm n ( t l'aj+l-a~ b)(t d t l,,,(t)) - 1 rim(-l) k ak+l+m-s n (s-a . ~ s ) d s

J ( 9 )

J =1 2ni =-la, J =l

fo r every m, k E IN.

To investigate the boundedness of the left term of (9) on G* we must

consider two different cases.

Let firsly t E Ge such that 1 t 1 rl. We can write

where denotes the argument of t. Notice . that

exp(+ 1 s 1 +.Bly)il fo r s=c+iy, yclR. By choosing reN such tha

( el ( 48, hence

r+ak+3+E>%+l and ak+3+r+m+e>a and by taking ~ a ( a ~ + ~ + m + r + c , a +m+r+&+l), one has

1 k+3

1 +m

s- 27r la

Let now

1 k+2

I ( ~ + i y - a ~ + ~ ) ( c + i y - a 1 I dy seW(m+r,k+2) s u p 1 j=i (s-aj)@(s)e61sl 1

t E G* and It 151. I t inrnediatelly follows f o r every m,k E H

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Moreover

k a l*a.i+l-ai C)(t ?g(t)) ctim

n ct d t j=l c-lco

We consider the paths of Figure 1

k+2 1 k+l

1

Figure 1

-Y

Since 1 im Ir t ak+l-s

r[ (s-a .)@(s)ds=l im ak+l-s n (s-a .)9(s)ds =O

y-m j =I Y+ m j =I J J

Y -Y

by applying the residues theorem we get

n (s-a .)@(s)ds = n (s-a .)@(s)ds + j -1 J J =1

J 1

+ 2ni Res { t %+l-s

r[ (s-a .)O(s); ~ = a ~ + ~ } J =1 J

By choosing .!EN such that e + l + ~ + % + ~ > % + ~ then

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ON K, TRANSFORMATION OF DISTRIBUTIONS

Also

lr+I t%+~-s n (s-a .)@(s)ds 1 s J =1 J

1

where a is a nonnegative integer such that c e [ a + ~ + % + ~ , a + ~ + l + ~ + ~ l . 1

By combining (10) and (11) we can conclude tha t M1 is a continuous 6 mapping from (18(an) into 9 ({an}:=l).

6 Finally according to I151, p.273, Ml*M(@)=@, for @ E 5 ({an):=l) and

M*M1(@)=@, for @ e U6(an) and the proof is complete.

3. AN AUXILIAR MAPPING .

This paragraph is devoted to analyze an auxiliar mapping that will be useful

in the next section.

Henceforth p>O , p G! H. Moreover here and in the following sections s-p 1 s+p 1 {an}:==l denotes the sequence of poles of the function RT + +

written in a such way that an+l<an, for every n=1,2, ... . A s i t is easy t o

1 ' + ;) are simples and i n f { ~ a , + ~ : see, the poles of IYT + k=1,2,. . . )>0.

Theorem 3 : Let 8 E [o,?). The mapping

is an isomorphism. 6 PROOF : Let @ e S (-m,m). Note firstly that (T @)(s) is a meromorphic

Ir function in cC that has a t most simple poles in s=an, n=1,2, ... .

To show that T is a continuous mapping we must analyze Ir

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186 J. A. BARRIOS AND J. J. BETANCOR

f o r every m, k E IN.

As it ' is easy t o see the function

has not poles in W(m,k).

On the other hand, from [7, 1(18(6)1

uniformly in x f o r x in any bounded interval. Hence there

real numbers M and Y such tha t

exist two positive

(13)

f o r s E W(m,k) and IIm s(>Y>l.

By defining

+ ~ ; ( r n , k ) = {s~W(m,k) : 1 Im sI<Y} and Wy(m.kI = {sEW(m,k) : I Im slzY},

we can wri te

But we have

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ON K, TRANSFORMATION OF DISTRIBUTIONS

k

. s !p ] exp( (*+:I I s 1 -B I 1-s 1 1, (s-a .I2 ~ - ( 3 n ) r ( ~ ~ ) r ( ~ ~ ) I 5

saWy(m,k J-1 J

for certain M1>O and r E W such that (1-s : saW(m,k)&V(r). 1 * +m+e+ Moreover, according to (13) by choosing n E IN such that n-- 2 %+l'

i t follows

k

s ~ ~ 1 s " n (s-a.)@(l-s)leBISI s M~ f sup le61slsn+'@(s)l J =1 J J = O seV(r)

for every s E ~ ; ( m , k ) and where M>O, i=2,3,4.

Hence

k

sup 1 exp((~+:) 1 s 1 ln IS-a .)(T @)(s) I s Y f sup 1 e61'1 sn*'~s) 1 (15) (m,k) J=l

J CI J = O seV(r)

From (14) and (15) we can deduce the continuity of T ~ . r '

We now define

2s+( 1/2)q(l-s) B+(rr/2)((a p~ = 3 p-S 3 p+S , f o r O € % n n=1

1 r(z + -)r(z - -1 2 2

Our purpose is t o prove that T is a continuous mapping from 6 P

~ * + ( ~ ' ~ ) ( { a ) into s (-a,a). n n=1

*+("")({a }a 1. Notice firstly (TpY)(s) is an entire Let E U n n=1 function.

Let n, m E IN. We choose k and r belongs to W such that %+l+csl-m, 3 km+m+- and a , + l + ~ + r + l r l + m > ~ + l + ~ Then 2

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188 J. A. BARRIOS AND J. J. BETANCOR

By (121, we can write

r for s E U W(t,k) and I Im s 1 >Y, where M>O and Y>1.

e=o By denoting again

fo r b O , ..., r, one has:

fo r a certain M1>O.

Moreover, by virtue of the restrictions of k and r and by (141,

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ON K, TRANSFORMATION OF DISTRIBUTIONS 189

k k

s u p 1 1 (s-a,)Ns)exp((e$) 1 s 1 ) 1s M sup I 1 (s-aj)~(s)exp((B$) 1s 1 ) 1 s ~ W ( 8 , k ) J=1 ' sEw(L ,~ ) J = I

with Mi>O, fo r i=2,3, and bO, l , . ..,r.

Therefore, there exist a positive constant M4 such that

and 7 is a continuous mapping from U 6 u

*+("/2)({a into s (-m,m). n n = l

Thus the proof i s finished because as i t is easy t o see T .T (@)=a, f o r 6 6 + ( ~ / 2 ) ( { ~ )m P P O E S (-m,m) and T .T (q)=Q!, f o r 9 E U n n-1

1. P P rn

4. THE CLASSICAL K -TRANSFORMATION. I-(

As i t was mentioned a t the begining of this paper the integral K -trans P

formation is defined by

where K denotes the modified Bessel function of third kind and order p. In P

this section we will investigate this integral transformation on some of the

spaces introduced in Section 2.

By virtue of well-known properties of the function K. (z ) (see P

[20l,p.173) i t can be seen tha t K,,{@)(y) is a holomorphic function for Re e E L

y>O, provided that E T (-m,m). We now introduce a new integral 6 4i

transformation over T (-m,m) denoted by K that extends K t o G6+(n/2). I-r I-r

6 4i Let # E T (-m,m) with 8 E lo,:), we define K {#Hy) by

P

oaexp( 1 K>#}(Y) = ymlJ fi K P (u)#lfldu , for y E G ~ + ( ~ / ~ ) (17)

0

where 1 p 1 <: and p d s a r g ySl*B.

Note f i rs t ly that if 1611<; and Bl+Sarg p Y i # , f o r i=1,2, then

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190 J. A. BARRIOS AND J. J. BETANCOR

according t o (3) and (4)

fo r every k e N, where MI, i=1,2, a re certain positive constants and rR i s

the path represented in Figure 2. Hence

l i m f r fi K (u))(gdu = l i m Ir fi K (u))(;)du = 0 R+w P R-JO P

and the integral in (17) is not depending on 7 provided that IBl/f and

6 Ssarg ysQl+Q. 1

heref fore, if y>O we can take Q1=0 and then

* Moreover, i t is not difficult to see that K {$)(y) is a holomorphic

P

function for y E G:+(~,~) dk *

and - K {g)(y) can be continuously extended t o dyk

G , for every k E N. 6 * The main property of K is established in the following

P * Theorem 4 : Let 6 E 10,;). p>O and p 6 El. K is an isomorphism from

P 6 T (-w,w) onto 5 Q+'K'2'(

PROOF : We will prove that

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ON K,, TRANSFORMATION OF DISTRIBUTIONS

Then by invoking Theorems 1, 2 and 3 the desired result will be established.

By virtue of (5) we get

-S s-(3/2) S-p 1 s+p 1 fi Klr(u) = 2 lYT + + T)ds , for Re u>O (18) c-100

with c>al. In effect, according to (12) i t follows

for Re u>O. Here 6" denotes the argument of u. From (19) it deduces that

is a holomorphic function for Re u>O. Moreover, fi K (u) is also holomorphic P

for Re u>O. Hence, since from (5) follows that (18) is true for u>O, (18)

holds for Re u>O. 6 Therefore, if 1 el 1 <: and ) e T (-m,w), then

The interchange of the order of integration is justified because the

integral

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1 92 J. A. BARRIOS AND J. J. BETANCOR

is absolutely convergent provided that 15 1 <:. On the other hand, by using the Cauchy integral formula we get

and we can conclude

y ~:+(n/2). * In other words KM{#}(y) = ( ~ - ' . ~ ~ . l l l ) { # } ( y ) , for Y e G n 2 . We now establish' a Parseval equation that will be useful in the sequel

79 Proposit idn 4 : If E T (-m,m), i=1,2, then

6 PROOF : If and #2 E T (-m,m), we can write

The intercange of the order of integration is justified because by virtue of

(3) and (4) i t follows

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ON K, TRANSFORMATION OF DISTRIBUTIONS 193

6 fo r a certain M>O, because r ( l + t p ) lq$(t) ldt < a , for every O E T ( - m , d .

0

5. THE GENERALIZED K -TRANSFORMATION. P

Let 6 E [o,;). For f E (56({an}~=l))', we define the generalized K P

transform K' f of f through P

3i 6 <K9f,@> = <f,K t$> , for each q5 E T (-m,m). P P

(21)

Notice that the definition (21) can be seen a s an extension of the

Parseval equation (20).

From Theorem 4 i t inmediately deduced the following

Theorem 5 : Let 6 E 0 W e generalized K' transformation is an P

6 t (n/2) isomorphism from ( 5 6

onto (T (-m,m)Ip. 6

On the other hand, by virtue of Proposition 3 if f E T (-m,m) then f 6

generates a regular distribution in (5 ({an}:=1))' by

6 Also, according t o Proposition 2 if g E 5 ({a 1 then g defines a regular

6 n n=1

distribution in (T (-m,m))' by

6 Therefoke, if f e T (-m,m) the

defined by

generalized K transform K'f of f is P Cc

6 , for 4 E T (-m,m)

and by invoking (201, (22) and (23)

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194 J. A. BARRIOS AND J. J. BETANCOR

l9 for Q, E T (-m,m).

Hence we can say that the classical K transform ia a special case of Cc

the generalized K transform. 12

Our purpose now is t o show that the generalized K transformation due Cc

t o A.H. Zemanian [19] defined by (2) is a special case of the generalized K P

transformation introduced herein. We need previously t o establish a relation

between the spaces X (see [I911 and f ({a }m- 1. P#a n n-1

Proposition 5 : Let a<O and O<p<l. 9l9({a }m ) is contained in X and n n=1 Cc~a

the inclusion is continuous. 6

PROOF : Notice firstly that if 0<p<1 the topology of Z) is

defined by the family of seminorms {rm,k}m,kc[N where

rm,o(Q,) = sup It m+p-(1/2) Q,(t)[ , for meH t€G6

for k even nonnegative integer and mcN, and

for k odd nonnegative integer and mcN.

On the other hand S =S Cc -P'

Therefore, for every kcH and Q, E Z)*({a 1 we get n n=l

and if a<O,

Thus, by taking into account that the topology of X i s generated by L a

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ON K, TRANSFORMATION OF DISTRIBUTIONS 195

{Pt,k}keN, the proof is finished.

From Proposition 5 i t infers that if f E X' then the restriction of f P'a

to 5*({a }m ) is in @@({a }m I)', when a<O and 0<p<1. Hence for every f E n n-1 n n=1 X' two generalized K transform of f can be defined. We now show that the

Ps a P generalized K transform of f E X ' given by (21) is equal (in the sense of

P P' a 6 equality in (T (-m,m))') to the generalized K transform of f a s given by

P (2).

Theorem 6 : let a<O and O<p<l. If f E X' then the generalized K w, a' P transform of f , a s defined by (211, is the regular generalized function in

6 (T (-m,co))' that is generated by the smooth function (K'f)(y) defined by P

(2).

PROOF : Let f E X' and denote F(y) = < f ( x ) , e K (xy)>, for Re y>a P'a P

and -n<arg y<n. According to A.H. Zemanian [201, Chap.VI, F(y) is a

holomorphic function for Re y>a and -n<arg y<lr and there exist a polynomial

P(y) and a positive constant M such that I ~ ( y ) l 5 ~ l / j ; (y-P+l)~(y), for

o<y<m.

Therefore by virtue of Proposition 2 F(y) defines a regular 6 distribution in (T (-m,m))' through

To show our theorem is sufficient to see that

p ( y ) + ( y ~ d y = < f ~ , K ~(xy))(y)dy>, for Q e T*(-~,.I. [ P

This statement can be proved by using Riemann sums and by proceeding a s in

the proof of Theorem 5.6-3 in 1201.

To finish this section we present an application of the generalized K P

transformation. We solve distributional differential equations of the form 6 P(S )u=g where P is a suitable polynomial and u and g a r e in 5

P Proposition 6 : Let O<p<l. The operator S defines a continuous linear

P mapping from 5*({an}y=l) into itself.

PROOF : By proceeding as in the proof of Proposition 5 i t is easy t o

see that for m, k E iN

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J. A. BARRIOS AND J. J. BETANCOR

6 Hence Sy is a continuous operator from 4 ({an):=l) into itself.

~ y ' virtue of Proposition 6 the generalized operator S defined on P

(4'({a la as follows n n=1

6 defines a continuous mapping from (5 ({an}~=l))' into itself. 6

Proposition 7 : Let BE[O,;). For every Q 6 T (-m,m),

PROOF : From well-known properties of the function K ( z ) (see [201, P

2 6 p.173) i t deduces that S K {Q}(x) = K {t $}(XI, for q5 E T (-a,=) and x>O. PtX P P

Hence the desired result follows. W

The following operation transform formula can be inmediately deduced

from Propositions 6 and 7.

Proposition 8 : Let O<p<l and 8~10,:). If f E a n ) and P is a

polynomial then

Let now P be a polynomial such that p(xZ) has not roots on G6 and

O<p<l.

We consider the differential equation

6 where g is a given member of (4 ({an}:sl))' and i t is unknown but also

required to be in (56({an)~=l))'.

By applying the generalized K -transformation t o (24) and invoking P

Proposition 8, we obtain

P ( x 2 ) ~ ( x ) = G(x)

where U and G are the K transforms of u and g respectively. P

Hence our solution for (24) is

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ON K, TRANSFORMATION OF DISTRIBUTIONS 197

2 6 because according to Proposition 1 1 R ( x ) is a multiplier in (T (-m,m))'.

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771-798, (1976). 4. J.B. Conway, Functions of one complex variable, Springer Verlag, New

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15. I.N. Sneddon, The use of integral t ransforms, Tata McGraw Hill, New

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