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=
=
8 ( l o g t)/t
butions (11.8.2) (11.8.3)
Now, it
E
E,,,u.
$(eX) eX> =
e
x
+
E
E
~
,
~
+
x
xI+(e 1 e > =
If we put $(x) = eSX with Re then (11.8.2) and (11.8.3) yield
s > a
and e ( x ) = esx with Re sew,

mspt  ILSP,,XI Re s > a, e (11.8.5) IMs Qt  ILs Q Re s < W. e Where IL denotes the Laplace transformation. virtue of (11.8.1) that (11.8.4)

Now, we deduce by
a < Re s < w. e With the help of this formula and the theorems of Section 8.3 of Chapter 8, we can get the results of the preceding Section 11.7.
(11.8.6)
MsVt
ILsPex
+
LsP
Remark. The use of the bilateral Laplace transformation (which
244
Chapter 11
is not studied in this book) yields, instead of (11.8.61, a simpler formula in which sLs does not occur and which does not require a decomposition into the form (11.8.1). On this topic, see Colombo C11 and Zemanian C31, Chapter 4. 11.9.Mellin and Fourier Transformations In this section the structure of a distribution defined in Section 11.3.and the results of Chapter 7 enable us to establish the following relations between Mellin and Fourier transformations. A distribution Vt and the interval I a l w C give rise to the family of distributions denoted(4) by e'rxV x and defined by e rx ,rl (11.9.1) <e V  x l $ ( x ) > =
,
,e(emx)> = at, tr'e(t), e"x where each $ ( X Iand each e ( x ) be such that tr'$(1og t) and tr'8(t) belong to the functions space on which Vt is defined. Now, we give the main result of this section. (11.9.2)
<el%
In order for Vt to be a Mellin transformable Theorem 11.9.1. distribution in the strip S a I w l it is necessary and sufficient that should be a tempered distribution (and r E 1 a , W[ , and er% ex therefore Fouriertransformable). If v(s) = MSVt, then we have (11.9.3)
v(r+2nic)
=
rc e"xV
5
x,
E
IR.
e Proof. Let p,q be finite such that a 5 p < r < q 2 w (with equalities being possible if a and w are finite). Let 6 and c 1 be such that 0 < < Sp and O < 5 ' < q 5. Put

r1 qt) = t $(log t) I t
(3.1.9.4)
$(XI If $ ( x )
E
' 0,
= e (rl)x$r(ex) , x
$ then we have that for all k
tk+lPC
(k)(t) @r
+
o
IR.
E
E
as t
IN +
tk+lq~'@~) It) + 0 as t + Therefore @,(t)
o+, m,
and hence it belongs to Ea belongs to E PI4 I
Mellin Transform
245
Conversely, if $,(t) E Ea,u, then $(XI E $. To see this, we consider their topologies and find that the spaces $ and E are isomorphic a,w by (11.9.4) and hence the first part of the theorem follows from (11.9.1). According to Theorem 11.6.2, v(r+2siC) can be majored by a m polynomial in 16 I as 151 + m. Hence the integral 1 v(r+2sic)$(E)d5 exists for all JI E $3. NQW, by virtue of the formui; (11.9.1) and the commutative property of the tensor product, we have trl 2nic
"XV

We shall see in Section 11.11 that the distributions erXV x e constitute a convolution algebra. 11.10.Inversion of the Mellin Transformation In the preceding sections we have derived the results of the Mellin transformation v(s) when the distribution Vt is prescribed. In this section, these results are considered in the inverse orientation, that is, we begin with some knowledge of v(s) and seek information about the distribution Vt' Theorem 11.10.1. If the function v(s) is holomorphic in the of finite width where s~+~v(s)is bounded for a certain strip S P?q integer K 2 0 as 191 + m , then there exists a unique distribution in E' called the antitransform (or inverse transform) of Mellin P rq of v(s) and denoted by IM;lv(s), such that (11.10.1)
m S milv(s)
=
v(s), p
<
Re s
Moreover, (11.10.2) and
K
tK g(t)
<
q.
Chapter 11
246
(11.10.3)
= (l)X(tD)K
lM;lv(s)
h(t)
where
and g ( t ) and h ( t ) are t a k e n e q u a l t o z e r o f o r t < 0. Before g i v i n g t h e proo€ of t h i s theorem, w e g i v e t h e f o l l o w i n g needed lemma. Lemma 11.10.1.If S
t h e f u n ct i o n F ( s ) i s holomorphic i n t h e s t r i p 2
(p,q a r e f i n i t e ) and i f s F ( s ) i s bounded i n S
PIq
Lim
[& (11.10.4)
then
F(s)
t > 0
t'ds,
f(t) = I
0,
t
i s t h e unique d i s t r i b u t i o n E E' with p < r PIq ? that is transform of F ( s ) for S Prq (11.10.5)
P?q'
r + i m
q which i s t h e a n t i 
Ex F ( s ) = f ( t ) .

Proof. By v i r t u e of t h e c o n d i t i o n s on F f s ) , f ( t ) e x i s t s every
where and i s independent of r . t h e n (11.10.4)
I f w e p u t t = eex and s = r+2niC;
gives f (eX) = erx
I
m
F(r+2niE) e
2nixs
m
d5
Hence (11.10.5')

e rxf (eX) = p x F ( r + 2 n i C ) .
w e have
A l s o , by t h e r e l a t i o n ( 1 1 . 9 . 3 )
(11.10.6)
F(r+ZniE) = IF erxf (eeX),
5
w e deduce from (11.10.5') t h a t 5' and it f o l l o w s by means of (11.10.6) and erxf(ex) belongs t o ,$; from t h e Theorem 1 1 . 9 . 1 t h a t f ( t ) belongs t o El and F ( s ) is t h e PI9 M e l l i n t r a n s f o r m a t i o n of f ( t ) Since F(r+ZniC) belongs t o $
.
Now w e show t h e uniqueness of f ( t ) . suppose t h a t Wte E' f o r any I M s W = F ( s ) PIq (11.10.7)
lM Y = IM S
w
s t

lMsf(t)
For t h i s purpose, l e t us
.
.
P u t Yt = Wtf (t) Then
Mellin Transform
247
which g i v e s (11.10.7
')
l M s Y = 0 on S
p,q'
This implies Y = 0 i n E' Now, by Theorem 1 1 . 9 . 1 , (11.10.7) P,9* implies t h a t 1 F e ' l x Y x = 0. Therefore, erXY x = 0 i n $ I . Hence e e Yt = 0 i n E ' by v i r t u e of (11.9.1) and t h e explanations given i n PI9 t h e Section 11.9. Hence w e conclude t h a t l M s W = F ( s )
.
Proof (of t h e theorem 11.10.1). L e t r ' be a r e a l number e x t e r i o r 1 t o Cp,q]. P u t F ( s ) = v ( s ) ( ~ + r ' )  ~A.l s o , lMt F ( s ) e x i s t s according t o Lemma 11.10.1 and i s equal t o f ( t ) . Now, by t h e r u l e (11.7.9) w e have s J F ( s ) = (1)'
lMs ( t D ) j f ( t ) ;
hence by inversion, IMlsJF(s) t
= (1)'(tD)Jf(t).
Since F ( s ) = v ( s ) / ( s + r ' I K , w e have v ( s ) = ( s + r l ) K F ( s )=
K
1
j=O
(g)rvK1sjF(s);
hence
K
1 (1)1(j)r S K  j ( t D ) j f( t ) . j0 1 This proves t h e e x i s t e n c e of t h e antitransform IMt v ( s ) , and t h e f i r s t p a r t of t h e theorem follows. =
, and does n o t S, denote t h e s t r i p A l s o , l e t c o n t a i n any of t h e numbers 1,2,...,K. i n t h e complex plane such t h a t R e s B 0 , and put G ( s ) = ~ ( s ) / ( s  K ) ~ . Now, by lemma 11.10.1, G ( s ) has t h e antitransform q l t ) . Consequently, by t h e r u l e (11.7.7), we have L e t Q be an open i n t e r v a l contained i n ]p,q[
K K K (1) YsD t g ( t ) = (sK), G ( s ) = v ( s ) on S,, hence K K K (1) D t g ( t ) = lMilv(s) which proves ( 1 1 . 1 0 . 2 )
.
Chapter 11
248
Similarly, formula (11.10.3) can be established with the help Also, the rule (11.7.6) can lead to a simple of rule (11.7.9). inversion formula.

Remark. In the above theorem, if p has no lower limit, then 1 1 lMt v(s) EEL,~; if q has no superior limit then IMt v(s) E EL,,; 1 v ( s ) these two conditions are simultaneously satisfied, then lMt E’ .v,m
.
if E
important remark on uniqueness. The above theorem states the uniqueness of lMi1v(s) with respect to a properly determined strip of holomorphia. But if the function v(s) is holomorphic in various strips parallel to the imaginary axis, which are separated by the singular points, and in these strips v(s) is majorized by a power of lsl,then there exist a s many distinct antitransforms 1 .v(s) as strips. A l s o , when we say the antitransform of a lMt function v(s), one should take care to choose the strip in which it is holomorphic. An
Problem 11.10.1
(i)
Eli1( s + z )  ’
=
U(tl)tz for the halfplane Re s
(ii)
lMil (s+z)’’
=
U(l;t)tz for the halfplane Re s > Re z;
(in) lM;’r(A)(sz)’
= Fp U(l;t)tzllog tl’”,
(iv) IMi’r(A)(sz)’
=
(v)
mi1
s’(l+s)l
ei’’Fp
<
Re 2
;
Re s > Re z;
U(tl)tz(log t)‘“,
Re
s
<
Re z ;
= u(1;t) (t1), Re s > 0;
s’ (l+s)’ = U(1;t)t (vi) IM;’
+ U(t11,
1
<
Re s <
0;
(vii) lM~lsl(l+s)l = (tl)U(tl) , Re s < 1; (viii) mt 1 (s21)’ = U(1;t) sin llog t[ , Re s > 0; (ix)
(s21)’
= U(t1) sin log t, Re s < 0.
Remarks. If Re s > 0, (11.10.4) does not exist for t = 0. For t < 0, f(t) = 0 due to extension, and we will not obtain a continuous function at the origin. This is the case of ( v ) which is discontinuous at the origin. If Re s < 0, the integral in (11.10.4) has a value for t = 0, and we obtain an everywhere continuous function f(t) which are the cases of (vi) and (vii).
249
Mellin Transform
11.11. The Mellin Convolution We have already seen the convolution of Fourier and Laplace transformations in Chapters 7 and 8. In this section we show another type of convolution (which we shall call Mellin type) that can be readily analysed by means of the Mellin transformation in the following manner. Definition 11.11.1.
Let V
E'
pl'ql' E'p2 '92 ? P = SUP(Pl'P2) and q = inf (qlIq2)where p < q. Then the Mellin convolution W \ V is the distribution belonging to E' defined by P?q (11.11.1)
<(W\ V),,$(t)>
E
= < WU,"Wt,$(ut)>l>, Y $ e E
P'q'
To prove the existence of (11.11.1) it is sufficient to show C E when u > 0. that W ( U ) =
7
tk+l$(k) (t) = k
p 'I, q+ 'I' (t) bk(t).
Put
Similarly, we can show that there exists< > 0 such that
Chapter 11
250
Hence w (u)EE
P19'
As an illustration of these ideas, consider the following. 11.11.1. Examples and particular cases 1. By making use of (11.11.1), we have
which yields the identity (11.11.2)
at\6(ta) ,o(t)> =
aU,[<6(ta),+(tu)>l> 1
=
Consequently, we obtain (11.11.3) 2.
1 vtL6 (ta) = ; vtIa,
By (ll.ll.l),
(a > 0).
we have
1 Y f (vLf)u = at,rf(U/t)>,
E
(1)
which yields VLf
E
ID' (f)(5).
If g is a locally summable function, then we have
Theorem 11.11.1. The Mellin convolution is associative and commutative (see Section 11.11.2). Theorem 11.11.2. The space E' is an algebra whose multiplicaPI9 tion law is the Mellin convolution and whose unit element is 6(t1). Proof. This is a consequence of Theorem 11.11.1 and formula (11.11.2) because from the definition 11.11.1, we have W\V
E
EiIq and W
E
E'
P,9'
Mellin Transform
11.11.2.
2 51
Relation with the Mellin transformation
With the notations being the same as that of Definition 11.11.1, let M s V = v(s) and lMsW = w(s). Then, we have
mS CWLV) = w(s)v(s) , p
(11.11.4)
<
Re s
<
q
.
which can be obtained easily by putting +(t) = t,s1 This relation can be related with the relations which exist between the ordinary convolution and the transforms of Fourier as well as Laplace. Problem 11.11.1. (1) Prove that for k
IN
vLs(k) (t1) = Dktkvt
(i)
(ii) G(ta)\ (iii) Vt\ (2)
E
6(ta') = G(ta.a'), a and a' > 0,
t 6'(t1) = tDVt.
Prove that
11.11.3.
Relation with the ordinary convolution
Let V and W
E
EAIw.
Then P = W L V
E
EA
.
Let $ ( x ) E $. Then according to the statements given in r1 $(log t) E Ea for r being a Section 11.9, we have +,(t) = t r a fixed number in the interval l a , w [ Moreover, by (11.11.1) we have
.

at,t
r1 +(log t)> = <wU
r1 $(log t ,P ht,t
log u ) > l >
Now, according to the statements given in Definition 11.11.1, we have
Hence, $,(y) which belongs to Ea ,W. given in the proof of Theorem 11.9.1.
E
$, according to the statements
Further, by making use of
Chapter 11
252
Section 5.9 of Chapter 5, (11.11.5) can be rewritten as (11.11.6)
<e'rxp
by virtue of (5.8.3)
 x , $ ( x ) > = <e'=YW , c <erXv ,,$(x+y)>~> e ee = < ( e  3 _,~*(e~~v x,$(x) > e e of Chapter 5.
The existence of the first member of (11.11.6) assures us that the convolution exists and hence we conclude
which illustrates the reason for calling the operation\ defined in Definition 11.11.1 the Mellin convolution. 11.11.4. The operator
"
(tD)
According to (iii) of Problem 11.11.1, we have tDV = [: t S' (t1)1\ V. By repetition, we further have (tD)2V = [ t&'(t1) I\
k&'(tl)l
LV
= [t&'(tl)J12\V=K2\V
where K2 = (tG'(t1)) L2 we obtain (11.11.8)
.
Iterating this (n1) times yields,
(tD)"V = [ t S'(t1) A n L V
=
Kn\ V
where
.
Kn = (t b'(t1) L n
i.e. K is the nth power of Mellin COnVOlUtiOn of t 6'(t1)* n Ws t 6' (t1) =  s I then (11.11.4) transforms (11.11.8) to (11.11.9)
ms (tD)"V
= (l)n S~V(S)
in accordance with (11.7.9). (tD)"V
=
Also,
(1)"(m;'Sn)
we deduce from (11.11.9)
\ V.
Further, we generalize this process by putting
Since,
M e l l i n Transform
253
(11.11.10)
with
where e a c h v i s a r e a l or complex number.
# 0,1,2,3,...,
If A
we
have a c c o r d i n g t o ( i v ) and ( v ) of Problem 11.5.1,
KX = FpU(t1) r ( 1 A
in E ' ,~ .
( l o g t )A1
A
I t f o l l o w s t h a t when V c Eh
w i t h a < 0 < w, t h e n ( t D ) V i s r e p r e s e n t e d by t h e c o u p l e K t \ V and K ; \  v ~ When o a < w, ( t D ) =
KlLV.
x
And when a < w 5 0 , ( t D ) V = IZ1
I f A i s r e a l , t h e n ( t D ) 'V only i f V
E
E'
a,w
w i l l be
real
'v
V.
(and e q u a l t o KX\
V)
w i t h a 5 0 , which o c c u r s i n p a r t i c u l a r ; when V is
r e p r e s e n t e d by a f u n c t i o n b e l o n g i n g t o I D ( 1 ) .
(For t h e c a l c u l a t i o n ,
see p a r t i c u l a r cases of S e c t i o n 11.11.1.) S i m i l a r l y , one can g e n e r a l i z e t h e o p e r a t i o n ( D t ) 'and t h a t t h e formula (1 1 . 7 . 1 0 ) s u g g e s t s t h e d e f i n i t i o n = eivn
(Dt)%
11.12.
[m;'(si)"
note again
J\ v.
Abelian Theorems I n s e c t i o n 1 1 . 4 w e have i n t r o d u c e d t h e a b s c i s s a e o f e x i s t e n c e
of v(s)=IMsV, a b s c i s s a e t h a t w e d e n o t e by a and u which a r e l i m i t e d by t h e w i d e s t s t r i p S a
i n which v ( s ) i s holomorphic.
Hence, i f an
I W
a b s c i s s a of e x i s t e n c e i s f i n i t e , t h e n t h i s i s t h e r e a l p a r t of t h e a f f i x of a s i n g u l a r p o i n t f o r t h e f u n c t i o n v ( s ) . L e t s = A and s * 2 be t h e s i n g u l a r p o i n t s c o r r e s p o n d i n g ( 6 ) t o a and W . W e now show t h e b e h a v i o u r of v ( s ) i n a neighbourhood o f t h e s e p o i n t s and c a l l t h e r e s u l t s of t h i s b e h a v i o u r a s Abelian theorems f o r t h e M e l l i n t r a n s f o r m a t i o n . The r e s u l t s p r e s e n t e d h e r e i n are q u i t e e q u i v a l e n t as i n d i c a t e d i n Lavoine and Misra [ 4 Theorem 1 1 . 1 2 . 1
1
[for t h e i n f e r i o r abscissa).
equal t o t  A \ l o g tIV[: H + h ( t ) + g ( t ) ]
I f Vt
c E'
a,w
is
254
Chapter 11
on 10,TC (if
,T
<
l/e, where
A,H,V are numbers such that Re
and Re v > 1,
A = a
(ii) h(t) is a function tending to 0 as t
+
0+,
(iii) g(t) is continuous function such that T
l+i Im(sA)dtl < M g(t) e T'
IJ
with M being independent of T' and s when 0 then (11.12.1)
lMsVt
 Hr(v+l) (sA)v1
as s
f+
E
+
A, with
2 arg(s~)2
4j 
Proof. Here the distribution P x e
c,
<
TI < T and lsAI < n,
E
> 0.
defined in Section 11.8 is
equal to
[.
eAx xv [ H+h (eX)+$ex)lon]( log t I ,
Section 8.11.2 of Chapter 8 is applicable here and the formula (8.11.8) of Chapter 8 and (11.8.6) give (11.12.1). Theorem 11.12.2 is equal to t'(log
(for the superior abscissa). If Vt€ El
aru
t)'
CHl+hl(t)l + gl(t)
on ]TII[ ,T1 > e, where HII v are numbers such that Re n =
w
and Re v > 1,
(ii) hl(t) is a function tending to 0 as t
+
m,
(i)
0,
(iii) g1 (t) is a continuous function such that .L
with M being independent of T' and s when T' > T1 and Is+nl < n, then (11.12.2)
MsVt
as s
f+
+
n, with
E
 H~ r(v+i) (ns) v1 3* 5 arg(sn) 5 2
€*
Proof. The proof is similar to that of the previous theorem but instead of P x we consider the distribution Q defined in Section 11.8. e e
Mellin Transform
255
In the following two theorems we now show the behaviour of v(s) at infinity. Theorem 11.12.3 (for Re s + a ) . Let Pt E E' be such that in a?the sense of Section 11.3.2 , Pt = Dk f(t), with the function f(t) satisfying the conditions: (i)
f(t) has its support in Cola] and
a
belongs to this support,
(ii) tamkf(t) is summable, H(log a/t)" as t (iii) f(t) Re v > 1. Then (11.12.3)
as s
+
m
mSpt
(1)k H r ( v + l ) ask sk'"
in an angle where larg
Proof. 
with w(t)

f
a0, where H is a number and
+
81
5
2
E.
We set
0 as t + a0.
Hence
where w(aex)
+
o
as x
+
o+.
Now, by virtue of the Theorem 8.11.1 of Chapter 8, we have a m m f (t) = I f (t)ts'dt = as! f (aex)e'ax dx S
0
S
= a
0
~ ~ ~ f ( a e H~ r)( u + l ) ass''
as s + w in an angle where larg s I 2 (11.12.3) by means of (11.7.6).
Qt
=
(i)
~
5
E.
Finally, we deduce
Theorem 11.12.4 (for Re s +  a ) . Let Qt E Elmlube such that Dk fl(t), with the function fl(t) satisfying the conditions: fl(t) has its support in [bra[ with b>O belonging to this supportI
(ii) twkfl(t) is summable,
(iii) fl(t) H(log t/b)v as t+b0, where H is a number and Re v >  l .
Chapter 11
256
Then (11.12.4) as
s +

$t
in an angle where
E
3n
5 arg s 2

E.
It is necessary here that if s is the halfplane Re s
as I s 1 + m , with a > 0, k complex) G independent of (11.12.5)
1 lMt v(s)
If the function v ( s ) is a and satisfies
lN, Re X 2 2, and number (real or
E
s,
k
= D
then we have p(t).
where p(t) is a continuous function for t > 0 which is null for t > a and such that (11.12.6) (1)kv ( s + k ) and wl(s) = asw(s) with Proof. We set w(s) s(s+l) ,....,(s+kl), k 1,2,3, .... In this setting wl(s) is =
( s ) ~=
( s ) ~
=
holomorphic in the halfplane Re s > sup(0,cik) and satisfies (11.12.7)
wl(s)
. (a)kG
s 1 as
1.
+

in this half plane. Now by lemma 11.10.1, we conclude that lM;lwl(s) is a continuous function p,(t) for t > 0. Moreover, by (11.12.7), we have for real r
Mellin Transform
257
This can be rewritten as 1 m (11.12.8) 1 P,(t) tr'dt + p,(t) tr'dt + 0 as 0 1 In (11.12.81, the first integral tends to zero as r + t > 1, tr1 grows with r. Hence, (11.12.8) requires
1:

+
m.
and when
pl(t) = 0 for t > 1. On the other hand, by (11.8.6) and making use of (11.12.7) we have
I L ~ P ~ ( ~= mspl(t) ~)
=
wl(s)
. (a)kG sAas
+ m.
Hence, by a Tauberian theorem well known for Laplace transformation (see Theorem 8.12.1 of Chapter 8) we have k pl(ex) x A  l as x + o+. ~
,w
It follows that
NOW, we put p(t) = pl(tla). Since w(s) = aSwl(s) and k v(s) = (1) (sk)k we have by the rules of Calculus (11.7.2) and 1 1 k (11.7.6) that Itw(s) = p(t), and finally, mt v(s) = D p(t) which proves the theorem. Theorem 11.12.6 (for Re s +  a ) . holomorphic in the halfplane Re s v(s) as
Is 1
+ m,
I
w
If the function v ( s ) is and satisfies
G bs(e'i"s)k'X
with b > 0, k
E
H I and Re s 2 2, then we have
where q(t) is a continuous function which is null for t which satisfies
<
b and
The proof of this theorem is similar to that of the previous theorem.
Chapter 11
258
11.13. Solution of Some Integral Equations In this section we shall derive briefly how the preceding theory of Mellin transforms may be used to determine the solution of certain integral equations. Consider the integral equation m
(11.13.1)
V(x)P(xt)dx
= Q(t), (t > 0).
0
The Mellin transformation of a distribution (Section 11.4) can be applied to solve such equations. For this purpose by applying the Mellin transform of a distribution on both sides of (11.13.1), we get
I
m
V(x)P(xt)dx
I
m
tSldt =
I
m
ts’Q(t)dt, 0 0 0 Taking y = xt and x as a variable in the left hand side of (11.13.2) and changing the order of integration by Fubini‘s theorem, we have (11.13.2)
where for V
E
EAlw,
IMs[P] = p(s) for P
L
EkIU,
MsCQI = q(S) for Q
E
EkIU#
IMs[V]
= V(S)
and their strip of definitions are represented by Sv, Sp, SQ, respectively. Here v(s), V ( x ) and Sv are unknown. The equation forces Sv to contain 1s as s belonging to a conventional subset of Sp n SQ. In other words, if pt denotes the set of s such that 1s E Sv, then the equation (11.13.3)
q ( s ) = v(ls)p(s)
%n
Sp n
holds for s
E
Replacing
by (1s) on both sides of (11.13.3), we have
Put k ( s )
8
1 P( s
== j
SQ.
and IMs[K(x)]
=
k(s)
=
1 p o. Therefore
Mellin Transform
259
q(1s) , then we have IMsBt = q(1s) . If Bt = ';MI (11.11.4) we have IMs (B\
By making use of
K) = q(ls)k(s)
and by (11.13.4)? M s ( B L K) = v(s) = IM V which gives S
(11.13.5)
V(X) = ( B I K)x.
Since q(s) = NsQ(t), we have Bt = t1Q(,)1; and by (11.11.4) and (11.13.5) we get V(x) =
IQ(y)1K(;)y
0
2 dy.
Now by putting t = I in above integral, we finally obtain Y (11.13.6)
V(x) =
m
I
0
Q(t)K(xt)dt
provided, of course? the inverse Mellin transform ~(x) = i.e.
(11.13.6)
mi1
[pol 1
,s
+
x) exists;
is the solution of integral equation (11.13.1).
In particular, the equation (11.13.1) will have the solution V(x) =
I
m
0
Q(t)P(xt)dt
if (11.13.7)
P(S)P(ls)
=
1;
that is, the equation (11.13.7) is a necessary condition of p to be a Fourier kernel (see Colombo C11).
+
Exam le. We mention below an example of such an equation. Take P(x) = x Y,(x) where Yv(x) is the Bessel function of the second kind
of order
v,
Then (see Sneddon 121, Problem 2.37 (b)) we have
s1/2 r(1/4 + s/2 + v/2) p(s) = 2 r (3/4  s/2 + v/2) and hence
so
that
cot(3" 4
2
+
Vfl 2)
260
Chapter 11
Now making u s e of Sneddon C2 1, Problem 2.38
( b ) w e see t h a t
where Hv i s t h e s t r u v e f u n c t i o n . I n o t h e r words, w e have shown t h a t t h e i n t e g r a l e q u a t i o n m
( x t ) % ( x ) Yv(xt)dx = Q ( t ) 0
has t h e s o l u t i o n
m( x t ) 4Q ( t H ) ,,(tx)dt.
V(x) = 0
Now, w e can d e r i v e t h e s o l u t i o n of t h e i n t e g r a l e q u a t i o n
i n a s i m i l a r manner. If w e c o n s i d e r W,R and G t o be i n E' , t h e n atw by (2) of Pro l e m 11.11.1 w e can w r i t e (11.13.8) i n t h e form (11.13.9) By v i r t u e of
11.11.4)
we obtain
(11.13.10)
where
and t h e i r s t r i p of d e f i n i t i o n s are r e p r e s e n t e d by Sw, SR and SG Here R , r ( s ) and SR a r e unknown. S i n c e s E Sw, respectively. sR and SG, w e can s a y t h a t (11.13.10) h o l d s f o r s E Sw n SR n SG. The e q u a t i o n (11.13.10)
can be w r i t t e n as
Mellin Transform conventional function. then (11.11.4) yields
261
If Gl(t) = lMt 1gl(s) and H(t) = lMilh(s),
R(t) =(GIL HIt. Now, by virtue of ( 2 ) of Problem 11.11.1, the above can be written under the form of the integral m
(11.13.12)
R(t) = !G,(y)H(t/y)y'dt. 0
The integral equation 1 (11.13.13) I Wl(t/x)Rl(x)rdx = Gl(t), 0 t
<
t < 1, W1,Rland G 1 ~ E h , w ,
is a more interesting form of the above equation. substitution
If we make the
R(t) = Rl(t) [U(t)U(lt)],
G(t) = Gl(t) [U(t)U(1t)]
W(t) = Wl(t)
where
[
U(t)U(1t)l
[,
and
O < t < l
U(t)U(lt) =
elsewhere,
then we see that (11.13.13)
is equivalent to (11.13.8)
and hence
to (11.13.10). 11.14. EulerCauchy Differential Equations
In this section we also illustrate the use of Mellin transformation in the following differential equations in a distributional setting. Let Xt be a distribution having support in ]O,[ satisfying N (11.14.1) 1 AntnDnXt = Vt n=0 where Vt is a distribution whose Mellin transform is v(s) in the strip Sv (see Section 11.4) and An # 0. Such equations are called at times EulerCauchy differential equations. The Mellin transformation generates an operational calculus by means of which (11.14.1) may be solved for the unknown X when Vt is a known Mellin transformt able distribution.

Put x(s) = IMs (11.7.8) ,we have
[
Xtl
. According to the Section 11.7
(formula
Chapter 11
262
and (11.14.1) (11.14.2)
transforms to P,(S)X(S)
= v(s)
where
Hence
(N, be the roots (real or complex) of Let ak, k = 1,2,3,...,K the polynomial PN(s) and let mk be their orders of multiplicity. If we suppose that a corresponds to the root which has largest real part. Then, l/PN(s) can be written in the form:
A l s o , we have
(11.14.4)
mi1 (sak)j = IL (j,ak ;t), Re s
>
Re a where
as can be seen in Section 11.5, formula (iv) of Problem 11.5.1. (See also Colombo and Lavoine [11 , p. 152.) The inversion of (11.14.3) by means of Mellin convolution yields, (see Section 11.11.4)
provided that Sv contains a substrip in which Re s z ak. Consequently (11.14.5) is a solution of (11.14.1). If Sv does not contain any s such that Re s > ak. Then the case is more complicated. Suppose SV be the strip a < Re s < w and let the roots be arranged such that Re al < Re a2 Re a3,. , Then, we have
..
(11.14.6)
1 IMt (sa)j = E(j,a;t), Re
s <
Re a,
Mellin Transform
where
IL(j,a;t)
=
263
(1)1 U(ta)ta ~T (7!
log1lt, j = 1,2,3
,...
If Re aKl < w < Re a by (v) of Problem 11.5.1. K'+1' then the inversion of (11.14.3) yields K' mk K mk (11.14.7) Xt = 1 1 B . V L JL(j,ak;t) + 1 k=l lk k=K'+1 j=1 where the B 's can be obtained by decomposition of l/PN(x) jk Particular case. Taking N = 1, (11.14.1) reduces to AltDXt
+
AoXt = Vt or
tDXt
+
1 Xt = V A1 t' A1 AO
That is
+
tDXt
(11.14.8)
AXt = Vt
1 by denoting AO V t by Vt' A,I by A and A, I form x(s) =
(11.14.9)
If
w >
Re A, (11.14.5) Xt =
(11.14.10)
and if
w
<
(11.14.11)
(11.14.3)
will now take the
v(s) SA gives
vt\
IL l,A;t),
Re A, (11.14.7) yields, Xt =
vt\
(1,A;t).
On the other hand if Vt is represented by a function h(t) , then (11.14.10) gives having support [ f.! ,y] contained in [ 0,[ Y
B < t < y. B If y = +m and if w < ReA, we have by (11.14.11) t Xt = tA f h(u)uA'du, t > B. (11.14.13) (11.14.12)
Xt = +A
f h(u)uA'du,
B
It can be easily verified that (11.14.12) and (11.14.13) give If v(s) is such that the solutions of (11.14.8). (11.14.14)
V(S) =
then (11.14.9) gives
(sA)g(s)
Chapter 11
264
(11.14.15)
X t
Xt =
1 mt g(s).
We remark here that (11.14.14) implies that the existence of such that V
t = tDGt
and the equation (11.14.8)
 AGtr
msG =
g(S)
,
can be written as
+
tD(Xt+Gt)
A(Xt+Gt) = 0.
Hence the solution Xt = G is obtained which is identical to t (11.14.15). If (11.14.13) and (11.14.12) do not give computable results, then one can consider (11.14.9) in the form of a series
Hence by (11.7.9) we have m
t'


1
n=O
(l)n Anl(tD)%t
under the condition that the series converges. The solution of an EulerCauchy differential equation for functions can be obtained in a different but very similar way to that of distributions. For instance, we seek a function h(t) continuous on [ 0 , y l (y is bounded) such that
(11.14.16)
d t;ir h(t)
+ Ah(t)
=
f(t), t
E
[0
r y ]
where f(t) is a Mellin transformable function having support in LO, Yl (7). Denoting h(y) by W, and making use of (5.4.3) of Chapter 5, we have d and (11.14.16)
h(t) = Dh(t)
+
W6 (ty)h(O+)6 (t)
yields with th(O+)b(t) = 0
Mellin Transform
265
which is similar to equation (11.14.8). By putting H ( s ) = IMs h(t), F(s) = M s f (t), and applying the Mellin transformation to (11.14.16') we obtain
and its inversion is
where f(t) is similar to that given by (11.14.10) and
1
x(O,y;t) =
0 I t 2 Y
elsewhere.
11.15.Potential Problems in Wedge Shaped Regions In this section we shall describe briefly how Mellin transformation in a distributional setting may be used to determine the solution of a physical problem which occurs in mathematical physics. We deal this work with a simple problem in potential theory. Consider an infinite two dimensional wedge as indicated in Figure 11.15.1. We choose a polar coordinate system u(r,e) with the origin at the apex of the wedge and the side of the wedge along the radial lines 8 =  a and 8 = a ( 0 < a < 2 r ) . Specially, the problem we wish to solve is the following: Find a function u(r,e) (which is a function of r and 8) in the interior of this wedge such that (i) it satisfies the partial differential equation
where 0 5 r 5 a and a 5 0 < a. The equation (11.15.1) 2 equation in polar coordinates multiplied by r ; (ii) it satisfies the boundary condition O z r l a (11.15.2)
u(r, +a)
=
r > a
(iii) u(r,e) is bounded as r is bounded.
is Laplace's
266
Chapter 11
Figure 11.15.1
To solve this problem we identify u(r,e) with a distribution in r. AS we see above that u(r,e) is defined only for 0 2 r 5 a and hence one can take u(r,e) = 0 for r > a, then (see Section 11.3) i.e. the Mellin transformation of u(r18) exists for u(r,e) E E; IRe s > 0. we may conclude that u(r,e) From the structure of u(r,8) E E; I is bounded as r is bounded. A l s o , u(r,+a) may be identified as U(a;t) and hence according to (11.3.6) we have
Consequently, we obtain Du(r,+a) =

6(ra) in Ei
,•
Hence, we may infer that u(r,B) satisfies the conditions (11.15.2) and (iii)
.
When applying the Mellin transformation we shall treat r as the independent variable and 6 as a fixed parameter: s1 M u(r,e) = (u(r,e), r > = u(s,O), Re s > 0. S
Now, by the operation transform formula
(11.7.8) of Section 11.7,
Mellin Transform
267
ms transforms (11.15.1) to

if we assume that a 2 can be interchanged with Ms. Therefore, we a e2 obtain (11.15.3)
U(s,e)
is8
= A(s) e
+
is8
B(s) e
where the unknown functions A ( s ) and B ( s ) do not depend upon 8. To determine A ( s ) and B ( s ) we first operate Mswith (11.15.2) and accordingly, we get
i.e. Thus, if MS [u(r,ta)] = U(s,ta) for then we obtain
s
E
Qu = Is: a < Re s < a 1, u1 u2
so that A(s) = B(s) =
2s
as cos s a
Consequently, (11.15.3) takes the form (11.15.4)
If
s =
a+iw, we have (since a <
0
< a)

Thus, we may conclude that s2 u ( s , 8 ) is bounded as 1. + and valid in the strip 71 < Re s < and hence one can say that U ( s , e ) is 4a 2a holomorphic in this strip. Consequently, U ( s , B ) in the strip 7F < Re s < satisfies all the needed conditions for the results 2a of Section 11.10. Thus upon invoking Theorem 11.10.1 with K = 0 or simply Lemma 11.10.1, we obtain our desired solutions:

268
Chapter 11
i.e. z4a. c h c 5. Thus, 2a according to Lemma 11.10.1, u(r,e) is the unique distribution in Consequently, we obtain the unique solution E;l, for 4 a < h < 2a. u(r,e) of our proposed problem.
and where h is a number lying between 71 4a
2a
11.16.Bibliography In addition to the works cited in the text, we mention the following references dealing with material of the present chapter. Fox c21, Fung Kang C11
I
Jeanquartier,P
[ 11.
Footnotes other authors call this the abscissae of convergence under the form of integral (11.4.2). One deduces the abscissae of existance directly form the structure of V, but it is easy to determine the singular points which fix the widest strip in which v ( s ) is holomorphic. Fp is not needed here if Re v $(s) =
r s
F
1.
and C is Euler's constant.
see Section 5.9 of Chapter 5. We employ this notation by recalling that if Vt = f(t) conventional function, we have e% x = erxf (em%). On the other hand e'rx6 (eX1)=6( x ) e Zemanian C 3 1 , p. 118.
.
several singular points may correspond to the same abscissa of existence as in the case of NsU(l;t) sin !log ti = (S21)' for Re s > 0. Such a plurality leads to introduce the functions g(t) and gl(t) in Theorem 11.12.1 and 11.12.2. a function of support [ O , y ] is Mellin transformable if its behaviour at the origin is known.
CHAPTER 12
HANKEL TRANSFORMATION AND BESSEL SERIES
Summary In recent years there has been considerable work on the use of Hankel transformation and Bessel series for functions in the solution of problems arising in mathematical physics. In this chapter we further develope this work in the distributional setting suitable for those whose interest lies in applications. For convenience we divide the chapter in two parts. The first part treats the Hankel transformation of functions in base spaces to distributions as in the definition of Fourier transformation in Chapter 7 and the formation of this distributional setting and its extension to several variables contain in Sections 12.2. to 12.1. Finally, we conclude this part by using this distributional setting of one variable in solving the heat conduction problem of circular cylinder. The second part deals with the work of Bessel series for generalized functions and the analysis of this topic including its application contains in Sections 12.9 to 12.17.
12.1. Hankel Transformation of Functions In this section we present those classical results of the Hankel transformation by means of Mellin transformation (see Section 11.1 of Chapter 11) which we need subsequently in the distributional setting of Hankel transformation. Let us take the function @(t) which satisfies the following conditions: (a) $(t) is defined for t
>
0;
(b) $(t)EL(O,m) (i.e. space of equivalence classes of functions that are Lebesgue integrable on ( 0 , ~ ) ) . 269
Chapter 12
270
If $(t) satisfies the above conditions, then we define the relation W
(12.1.1)
$(Y) = Mf:$(t)
=
1 $(t)
0
F t Jv,(yt)dt
1 is a real number, and J v denotes the Bessel where y > 0, v >  7 function of the first kind and of order v . In this relation we shall say that the function q(y) defined by (12.1.1) is the Hankel transform Of the function b(t). For brevity, we write Mvtransformation instead of Hankel transformation of order v.
A standard result concerning (12.1.1) inversion theorem.
is the following
Theorem 12.1.1. If $(t) E L(0,m) , $(t) is of bounded variation in a neighbourhood of the point t = y, and $(y) is defined by (12.1.1), then OJ
(12.l. 2)
b(t)
=
M;'&(Y)
= M ~ $ ( Y=)
1 m(y)Gt
0
Jv(yt)dy, 1 v 2 T'
where Mi1 denotes the inverse Hankel transformation. Here $(t) defined by (12.1.2) is called the inverse Hankel transform of $(y). 1 Note that, when v 2  7, this inverse Hankel transformation M i 1 is defined precisely the same formula as is the direct Hankel transformation mV ; in symbols, m,, = m~~
.
Now we establish the following result which will play an important role to obtain our main results.
then we want to prove
y
(y) =
$
.
(y)
We now prove our desired result. For this purpose, we recall that (see Erdelyi (Ed.) [a], V01.2, p. 227(7)) (12.l. 4 ) I
2s1
1 r(Tv+s + $/r
vs
(T+
3 $ ,
v > 1, s > 0,
where Ws denotes the Mellin transformation as indicated in Chapter 11.
Hankel Transform
271
Further, we impose the ccaditians (a) and (b) of Section 11.1 of sapter 11 of the
M e l l i n t r a n s f o r m a t i o n of f u n c t i o n s on $ and
.
4,
and s e t F(s)=IMs $ ( y )
and f ( s ) = lMs $ ( y ) Now w e have a c c o r d i n g t o (12.1.1) of F u b i n i ’ s theorem, m
m
F ( s ) = IMs$ =
J
yS’dy
0
0 by p u t t i n g y t = u. (12.1.5)
$ ( t ) q tJ v ( y t ) d t 0
m
=
and by means
m
1 u s1
ts$(t)dt
0
u4 J v ( u ) d u
Then w e have
F(s) = f ( l  s ) B v ( s ) .
Also, l e t g ( s ) = lMsy ( y ) . Then o p e r a t i n g s i m i l a r l y w i t h (12.1.3) t h e manner as e x p l a i n e d above, w e g e t
in
Hence, w e g e t a c c o r d i n g t o (12.1.5) g ( s ) = f ( s ) B v ( s ) By(ls). Because By(s) B V ( l  s ) = 1 (see Colombo C11 ) , w e have g f s f = f(s). Now,by t h e Theorem U.1.1 of C h a p t e r 11 of t h e M e l l i n t r a n s f o r m a t i o n w e g e t y (y) = $ (y). Therefore, we f i n a l l y obtain m
(12.1.6)
$ ( Y ) = M\ $ ( t ) =
$ ( t ) q tJ y ( y t ) d t , 0
and i t s i n v e r s i o n a c c o r d i n g t o (12.1.2)
is
(12.1.7) I f w e r e p l a c e t by y and y t o x i n t h e above i n t e g r a l , t h e n w e o b t a i n
(12.1.8) If w e r e p l a c e y I n t h e above i n t e g r a l t c a n a l s o be r e p l a c e d by x. by t and y t o x i n t h e i n t e g r a l of ( 1 2 . 1 . 6 ) , then w e g e t
(12.1.9)
$(XI
m
= EI;’$(y)
=
M :
$(t)&t Jv(xt)dt.
$(y) = 0
272
Also,
Chapter 12 in this integral, t can be replaced by y.
12.2. The Spaces Hv&
H;
Our study made on the spaces of base functions, generalized functions and distributions (see Chapters 2 to 5) enable us in this section to construct the spaces Hv and H; which provide the structure of a distribution to formulate the distributional setting of Hankel transformation in our subsequent work. Let I denote the open interval 10IC and v be a real number. By HV (denotes HV(x) whenever the variable needs to be specified) we denote the space of infinitely differentiable and complex valued functions $(x) defined on I such that (12.2.1) is finite for all nonnegative integers k and m. The space Hv is provided with a topology defined in the following manner : sequence { $ . I , j 7 and only if A
E
sNI
ld

converges to zero in H k
as j
+ m l
if

XEI
as j + for a set of nonnegative integers k and m. A l s o , the space HV is complete, (see Zemanian C31, Theorem 1.8.3). Lemma 12.2.1. $(x) is a member of Hv if and only if satisfies the following conditions: (i) $(x) is an infinitely differentiable and complex valued function on 0 < x < m; (ii) for each nonnegative integer k, (12.2.2)
$(XI =
x
+...+ aZkx2k + R2k(x)]
v+1/2 Cao+a2x 2
where the a's are constants given by (12.2.3)
a2k
1 7 1 k x~+L$(x) ' l i m (x D) k! 2
x+O+
and the remainder term RZk(x) satisfies
Hankel Transform 1D)k RZk(X) = O(1J x
(12.2.4)
(X
c
273 O+;
(iii) for each nonnegative integer k, Dk Q (x) is of rapid decrease as x c (i.e. Dk Q (x) tends to zero faster than any power of l/x as x + m ) .

Proof. Assume that $(x)
Condition (i) is satisfied by definition. The proof of (ii) and (iii) can be carried out as indicated in Zemanian [ 3 ] , pp. 130131. The conditions (i), (ii), (iii) imply that Q is in Hy , E
Hv.
Note that, for any fixed y L 1 , K y Jv(xy) as a function of x satisfies conditions (i) and (ii) of above lemma. However, it does not satisfy condition (iii) since
(See Jahnke, Emde and Losch is not a member of Hv.
The space H :
[ 11
, pp.
134 and 147.)
is the dual of Hv, and it is the space of
(H:(x))
distrlbutions(continuous linear functionals) on Hw.
V
E
H;
on Q
E
Hence Jxy Jv(xy)
The value of
Hy is usually denoted by
By I D ( 1 ) we denote the space of infinitely differentiable and complex valued functions $(x) with bounded support properly contained in I. The topology of this space is defined in the following manner: a sequence { $ . I , j 3 and only if
as j
F
m,
for each k
E
E
IN tends to zero in I D ( 1 ) as j
3 m,
if
IN.
Also, D(1)is a subspace of HV and convergence in D(I) implies convergence in Hv (see Zemanian 1 3 1 ) . Consequently, the restriction of any element of H : to ID(1)is an element of ID'(1) , the dual of D(1) and the space of distributions with support in I. By E ' ( 1 ) we denote the space of distributions having bounded support with respect to I. We remark here that the spaces defined
274
Chapter 12
herein bear the close resemblance to those of Zemanian C31. 12.3,Operations on H,, and H:, In this section we establish some important results of operations on H and H :
.
Multiplication h
For any real number h and v I the mapping $(x) + x $ ( x ) is an It follows that V + xhV defined by isomorphism of HV onto H v + h . A
(12.3.1)
<x v,
$(XI> =
is an isomorphism of H:+hon
a, xx
+(XI>
(For the proof see Zemanian C3] p.135.)
. : H
Operators We shall use the following differentiation operations: (12.3.2)
d 2v+l $ x~1/2)km skv = (xv1/2 sx I dx
(12.3.3)
d k x~1/2I Rk v = (x1 ad
where k is a nonnegative integer. The transpose of (12.3.2)
: = (xv1/2 Dx2v+l Dxv1/2 ) k in H
')
(12.3.2
and the transpose of (12.3.3
is
^k
')
.
Rt is
~1/2(Dxl) k
Rv = x
where as usual, D denotes the distributional derivative. Here we *k call S^k v and R v as transpose differential operators in the sense of Section 5.4 of Chapter 5. Let Mv
I
Nv
I
N i l be the operators on Hv defined by
(12.3.4)
Mv$(x) = x
(12.3.5)
N,,$(x)
=
~1/2 d
x
~+1/2+(~)
E X
v+1/2 d
azx ~  1 / 2 4 (XI
Hankel Transform
27 5
A
Further M and N are defined by
Thus, A
(12.3.7)
Mv = x
(12.3.8)
Nv = x
A
xv+1/2
v1/2D X
xv1/2
v+1/2D X
We summarize these results by: N M

N
V
V V
is an isomorphism of
HV
onto Hv+l
whose inverse is N;'
;
is a continuous linear mappinq from Hv+l into Hv; is a continuous linear mappine of H$ into H$+l;
A
M V is an isomorphism from H'v + l into H{
.
Moreover 4vL1 2 LI
A
,
.
M v Nv

2
Dx
4x
Also, we denote
':
, . a
= M~ Mv+l"'
'v+kl'
Note that we have the following equivalence relations between these operators:
ik
C
I
A
= (Mu N v ) k ;
A k k+v+1/2 Rv

^k Pv
.
We end this section by giving an important differentiation formula (12.3.9)
(see Koh [13
)
.
Chapter 12
276
12.4, Hankel Transformation of Distributions
By the results of Section 12.1 and the structure of a distribution described in the preceeding sections we formulate in this section the distributional setting of Hankel transformation by means of Hankel transformation Of functions in Hv to distributions in H : in the following manner. If Q E Hv, then it has according to equation (12.1.8) Hankel transformation of order v as
, the
m
(12.4.1)
4 (y)
and by (12.1.9)
we have
(12.4.2)
$(XI
By (12.4.1)
=
M : 4 (XI =
1
= Mv
0
4 (t)
Jv(yt)dt:
m
i(y) = lH:m(y)
and (12.4.2),
=
0
Q(t)
6Jv(yt)dt.
we can state the following results:
4 are functions of y and
x which belong to Hv;
1.
M u 4 and
2.
if a sequence {Qn} + 0 in the sense of Hv, then MvQn + 0 and M i 1 $n +. 0 in the sense of H,;
3.
in (12.4.1) and (12.4.2), Q(x) and $(y) are called the antitransforms (or inversetransforms) of 4 (y) and Q (x) respectively. Also,
by (12.4.1) and (12.4.2), Mt M;
$(XI
=
we may write
$(XI ;
M; MZ O ( Y ) = O(Y). Hence, we may conclude that M : or M : is self reciprocal. Therefore, are reciprocal automorphism of each other on H,: if and M :
Zit Q
E
H,
, then
Mv 4
E
H,
.
The above result permits us to make the following definition: The Hankel transformation of order v of a distribution V E H:(x) is the distribution belonging to H:(y) which is denoted by M , V (M:Vx whenever we desire to make the variable precise) and is defined by
Hankel Transform
277
This is a definition by transposition (Section 5.1 of Chapter 5). To conform with established terminology, we shall say that every generalized function (or distribution) belonging to H : is a Mytransformable generalized function (or distribution). A l s o , according to (12.4.2),
(12.4.3')
<M:v~,
4(x)>
M :
(12.4.3) is equivalent to =
(vxI~(x)>, for every
E
H".
Since MV4 belongs to Hv, we deduce from (12.4.3') that M uv belongs to H{ and hence the transformation defined by mv in (12.4.3) is an automorphism on . : H If the distribution V is associated with a summable function h(x) on 0 < x < , then (12.4.3) leads to the equality m
(12.4.4)
MVh(x)
=I h(t) flJv(yt)dt = 0
Mzh(x)
in accordance with (12.4.1). Thus, we may infer that the present setting of Hankel transformation of distributions properly generalizes the Hankel transformation of functions. 1
Examples. (i) If h(x) = x v1/2 (x+a) (12.4.4) that
.
Then,we have by
m
MVh(x)
=
I tv1/2(t+a)1 0
=
3.
c t Jv(yt)dt

avsec(vr)y1/2 [ ~  ~ ( a y )
ray)^
1 3 where T1 < Rev< 7, v # 7, larg a1 < r and Hv(ayI and Yv (ay) are the Struve function and the Bessel function of the second kind. (See Erdelyi (Ed.) [2] : V01.2, p. 22.)

(ii) Let h(x)
=
x' +'I2
(x2+a2)'I.
Then we have
m
Mvh(x) =
1 tv+1/2(t2+a2)1
F t Jv(yt)dt
0
= Irav1sec(vr)y1/2CIv(ay) 2
where Re a > 0,

1 Tv
<

~  ~ ( a Iy )
5 Iv and Lv are the modified Bessel Re v < z,
function of the first kind and the modified Struve function. Erdelyi (Ed.) [21 , Vol. 2, p. 23.)
(See
Chapter 12
Another particular case As we see in the proof of Lemma 12.2.1,
Jv(xy) as a
function of x does not belongs to Hv, and therefore the statement MvVx
(12.4.5)
=
XI
Jv(xy)>
However, under certain is not well defined for every V E H : . restrictions on V, (12.4.5) will possess a meaning and will agree For this purpose note that H; with the definition (12.4.3’). contains the distributions of the form: if Vx = Dk s(x) where s(x) is zero for x < 0, and locally summable for x > 0 and xa+1/2k+1s(x) is bounded as x + O+ for a A l s o , if there exists n > 0 such that x~s(x)is bounded as x +
c
.
v.
Then we have under these conditions:
ax, $ (x)>
(12.4.6)
klms(XI
= < s (x),(1)’$ ( k ) (x)> = (1)
$ (k)
(x)dx.
0
we now verify the existence of (12.4.6). Proof. Indeed, by our proposed structure of Vx, we have m
(12.4.7)
= (  1 ) k01 j xak+3’2s(x)
0
xka3/2$ (k)(XI dx
+ (1)k 1 xn~(x)x~$(~)(x)dx. 1 The second integral in (12.4.7) is bounded according to our A l s o , the hypothesis and the relation (12.2.3) of Lemma 12.2.1. third integral in (12.4.7) is bounded according to our hypothesis as well as condition (iii) of Lemma 12.2.1 and by relation (12.2.1). This proves (12.4.6).
If
v
has a bounded support contained in I, then M v V defined by
(12.4.8)
M V V = < VX ’ G J V ( x y ) >
,y
> 0.
This case will be discussed in detail in the Section 12.4.1.

Remark. The relation (12.4.3’) enables us to assign a Hankel transformation to distributions equal to certain increasing functions such as xn, n > 0, because $ (x) decreases more rapidly than every
Hankel Transform power of X1 as x

Examples.
+
279
.
If IL denotes the Laplace transformation, then show
that
xi:
(i) when V
X
ePxVx
IL
=
Jyx J"(YX)V~
is represented by a function $(x)
E H ~ .
Proof. We recall that Mt Q (y) =
~6 Jv (XY),$
(Y)>.
Now (12.4.3) gives 4
M :

e 'xVX,$(y)>
px x
=
for all
$
E
Hv.
e'p"v
X =
1 (ii) For v 1. 7 and 0
c < vx,G
J"(XY) e'Px>l,+(y)
>
where
u (xa)U (bx) =
c
< V x , G Jv(xy)ePX>
<
a < b
a+l
U (xa)U (bx) ax
Proof. 
$(Y)>
Consequently, we have M :
M yFP
my
= aI
Jxy
<
, show that
dx+L+l
J,(xy)& xa
1
i f a < x < b
0
elsewhere.
Jy(ay)
b GJ,(x) xa
U (xa)U (bx) = Fp fbL xa Jv.(xyjdx Fp xa a b J v (xy)d x + 6 J v (ay)log E] lim[ E+O a+€ a+1 (xy)dx +&J, (ay) log €1 lim E+O [a+€ I A&Jv xa
+
b
I a+l
Jxy
Jv(xy) lx xa
.
280
Chapter 12
By replacing log
E
by
a+l
 I
a+e U (xa)U (bx)a+l Fp xa a
=I
(iii) For v 2 (a) (b)
 z,1
xa dx, we obtain KYJ~ (XY)KYJ~ (ay)
show that in the sense of equality in H :
Mf: x2n+1/2 = v(v24)..
. (v24n2) y2n3/2,
x2n1/2= (v2i)(v 29).
M :
7
G J v (XY) dx . a+l xa
+
xa
, 2n,
..(v2 (2n1)2)y2n1/2
I
v > 2n1. where n is a positive integer. Proofs. 
From Magnus, Oberhettinger and Soni
[11, we have
If we put t=y, a=x and u=X+l, in this integral and consider this formula as a Hankel transformation, then we get
Now, apply M:to
both sides, we have
If X = 2n+11 we get
If X
=
2n, we get
Y 2n1/2 = (v21)~v 29)
M V
...(v2(2n1)2)y2n1/2
I
v 2n1. Remark. Since y does not belong to Hv(y) and hence (1) does not exist in the sense of the ordinary Hankel transformation.

12.4.1.
The Hankel transformation on 6’ (I)
As mentioned above, the Hankel transformation of certain (but not all) members of H : takes the form
(12.4.9)
b(y) = M w B x
=
Hankel Transform
281
W e s h a l l e s t a b l i s h t h a t when Bx E E' ( I ), b ( y ) i s a smooth f u n c t i o n on, 0 < y < m. I n d e e d , it can be e x t e n d e d i n t o an a n a l y t i c
f u n c t i o n on t h e complex p l a n e whose o n l y s i n g u l a r i t i e s a r e b r a n c h p o i n t s a t t h e o r i g i n and a t i n f i n i t y . To do t h i s , l e t z = c+ in be a complex v a r i a b l e , and s e t (12.4.10)
b ( z ) = IH; I f Bx
Theorem 1 2 . 4 . 1 .
BX = < B x 1 6 J v ( x z ) > . E
&'(I),
t h e n zv1/2b(z)
i s an e n t i r e
f u n c t i o n of t h e complex v a r i a b l e z ( i . e . it i s holomorphic i n t h e f i n i t e zplane)
.

E'(I),
t h e n i t s s u p p o r t is c o n t a i n e d i n t h e k i n t e r i o r of lo,[. A l s o , i f w e s e t B = D s ( x ) where s ( x ) i s a X c o n t i n u o u s f u n c t i o n having s u p p o r t i n [ a r i 3 l , 0 < a < 6 m. Then, by making u s e of t h i s s e t t i n g , w e have P r o o f . If Bx'
(12.4.11)
b ( z ) = (l)k 1 s ( x ) ?;[= dk Jv(zx)ldx.
a dx Making u s e of t h e series expansion of J ( z x ) (see Problem 1 . 4 . 1
of
C h a p t e r 1) w e have
or
where
Since
2'jj!u
j f i n i t e zplane.
. Remark
is bounded a s
j
f
m,
t h e series c o n v e r g e s i n t h e
T h i s p r o v e s o u r theorem. Since
b ( z ) z'li2
i s holomorphic i n t h e h a l f  p l a n e
y = R e z > 0 c a n b e s e e n above and c o n s e q u e n t l y w e may i n f e r t h a t b ( y ) i s a smooth f u n c t i o n on 0 < y < m. Theorem 12. 4. 2. d e f i n e d by ( 1 2 . 4 . 9 ) .
I f v 2 1/2 and i f L e t Bx E e'(1). Then, b ( y ) s a t i s f i e s t h e i n e q u a l i t y V + V 2
0 < y < 1
(12.4.12)
1 < y < 
Chapter 1 2
282
where K and p are sufficiently large real numbers.
C31,
Proof, The proof can be carried out as indicated in Zemanian 
pp. 1 4 6  1 4 7 .
1 2 . 5 . Some Rules
This section provides an account of the operational transform formulae for the spaces Hv and H:
.
12.5.1.
Transform formulae for Hv
If 9
E
Hv, we have
(12.5.1)
My+1(X9) =
(12.5.2)
mv +1(Nv9)
and if 4
E
 NvMv9;
= Y xv9;
Hv+l, then
Proof. The proofs of 
to ( 1 2 . 5 . 4 ) and ( 1 2 . 5 . 6 ) to The formula ( 1 2 . 5 . 5 ) f o l l o w s directly from ( 1 2 . 3 . 9 ) . To prove ( 1 2 . 5 . 8 ) we use the following recurrence relations: (12.5.1)
( 1 2 . 5 . 7 ) are given in Zemanian C31, pp. 1 3 9  1 4 0 .
(12.5.10)
Jvl(X)
+ Jv+l(X)
(see Sneddon C21, pp. 5 1 0  5 1 1 ) . Making use of (12.5.10) we have
=
2v x
Jv(X)j
Hankel Transform
283
m
=
Hence (12.5.8) is established. (12.5.10) we have R.H.S.
Of (12.5.9)
=
f
0
JV(xy)4 (XI
dx =
MY$(XI.
Further making use of (12.5.11) and m
1
dx
5 { 2 ~~[Jv,l(~y)Jv+l(Xy) ]X+(X) 0
m
= 1 1 2 v /2.J:(xy)x$
4v
12.5.2.
0
(x)
dx+2vj
1
0 xy
Transform formulae for H :
We now state a number of operationtransform formulae for the generalized Hankel transformation. These are exactly similar to the formulae of the preceding section, but deal with generalized operations. If V
E
Hi, we have

(12.5.12)
Mv+l(X.V) =
(12.5.13)
Mv+l(NVV)
= y
(12.5.14)
2 mV (X V)

=

NvMvV; MvV; 1
MvNvMvV;
284
Chapter 12
2
. . A
(12.5.15)
M v (MvNvV) = y M V V ;
(12.5.16)
Mv
and if V
E
H:+lI
cqvxl
= (1)k y2kIHvvx:
then *
(12.5.17)
MV (xV) = M v M V + l V ~
(12.5.18) (12.5.19)
M v (DxVI = gi(2v1) MV+1V(2~+1)Mv1V3..
Proof. The proofs of (12.5.12) to (12.5.15) and (12.5.17) to (12.5.18) can be found in Zemanian [ 3 1 1 143144. The formula pp.
(12.5.16) can be obtained by applying (12.5.15) successively k times k and making use of S*k v = (MvNv) which is the equivalence relation given in Section 12.3. The formula (12.5.19) can also be easily obtained by using recurrence relations (12.5.10) and (12.5.11). A
n
Problem 12.5.1 For k = OIl121....I show that
^k Vx = (1)k y2kM V V x I Vx
(i)
MvSV
'(ii)
M R
^k
v
v
vX
k
H :
k~1/2 Vxt Vx
= y M ~ + ~ x
*k k (iii) M v P v Vx = y Mv+kV x I Vx where
E
ikI ik and P*k
t
;
E
H : ;
H$+k
are defined in Section 12.3.
12.6. Inversion In Section 12.4 we have described the distributional setting of Hankel and inverse Hankel transformations. The present section further work out the inverse Hankel transformation by working with distributions in H : and we term this relation as an inversion of the Hankel transfornation of distributions. The main result of this section is: Theorem 12.6.1.
Let W Y
E
H$(y).
If Vx
E
H:(x)
and such that
285
Hankel Transform
(12.6.1)
= WY'
M:Vx
fo
then Vx i s c a l l e d t h e i n v e r s e Hankel t r a we obtain
vx
(12.6.2)
= M ' :
f W
Y
and con eque t l Y I
wy.
Proof. According t o (12.6.1)
I H :
m
and (12.4.3')
W , $ ( x ) > = cIH:
Mf: V X I J , ( x ) >
Y
=
<myv
VX'
w e have
I
XI;*(%)>
I
Ti J, E
Hy
= wx'm;M;*(X)s
= < V X I $(XI >
which y i e l d s t h e r e l a t i o n Mt W = Vx. Y
Hence (12.6.2)
i s established.
I n addition X
Y
because
The exchange formulae (12.6.1) and (12.6.2) enable u s t o c a l c u l a t e numerous transforms. We mention below a f e w examples of them. Examples. Show t h a t
(i)
M;
(ii) I H :
6(xa) =
Jax
Jay J v ( a y ) ,
J ~ ( ~ x =)
a > 0, y > 0;
6(ya);

Proofs. According t o (12.4.8)
IH; 6 (xa) = < 6 (xa) and hence (i) i s e s t a b l i s h e d ,
IH;
Jay
~ " ( a y i= 6 (xa)
I
w e have
,G J v ( x y )
7
=
Also, by (12.6.2),
& Jv,(ay).
Chapter 12
286
and by changing y to x, we have
which proves (ii).
Now by ( 1 2 . 4 . 8 1 ,
=
by (12.5.10).
12.6.1.
we have
 @ Jy(ay)  & a u1 (ay) +$
Jy+,(ay)
This proves (iii),
Remarks The space H:
does not contain the Dirac functional 6(x) nor its
derivatives. If we take a semiclosed interval [ O , w [ : in place of I But in then 6(x) and certain of its derivatives would be in H: this case each of the elements of this space would not have a unique transform nor a unique inverse because the transform of 6 ( x ) then is null and hence Mywould not be an automorphism on H: Let us illustrate this remark with the help of the following example.
.
.
1.
According to (12.4.8) IH;~(X)
= <6(x)
, we
have
Jxy, J"(xY)>
= 0,
1
v >
 2'
If M : Vx = W then we may write IHz [vx+c6 ( x ) 1 = W for any c. But Yr Y in this case, the inverse of W , according to Theorem 1 2 . 6 . 1 , is not Y unique and it 'would be Vx+c6 (x)
.
Moreover, in certain cases we see that the distributions in H ; associated with 6 (k)(x) For instance, the equation XX = 0 has the solution X = 6(x) in ID' and the solution X = F in Hd defined by
.
lim x'"'1/2#I(x)
,
Q
E
H ,
;
X'O+
because <xF,Q(x)> =
meniber of H:
(x) is finite.
That this truly defines F as a
follows directly from the condition (ii) of Lemma 12.2.1.
Hankel Transform (See Zemanian
[a],
287
pp. 151152.)
I In the preceding case we have taken v 1.  2. However Zemanian [ S ] has shown that a Hankel transformation of any order can be defined. Briefly, we see this as follows.
2.
Let v be a real number and let k be a nonnegative integer 1 such that v+k 1.  z for every $(y) E Hv We now set
.
Let Vx E H'(x) and if the Hankel transformation of order v+k is denoted by M:Vx which is a distribution in Hl Then X M ' V can be defined as v x
.
of V
<M:Vx,$(X)'
=
Mv,k$(X)>

1 in M:, then M: for every Q E Hv. If we take v 1 7 = M y and it follows that M; Vx coincides with the Hankel transformation of distributions defined by (12.4.3). 12.7,The nDimensional Hankel Transformation In the preceding sections we have extended the Hankel transformation to certain generalized functions of one dimension. In the present section we develope the ndimensional case corresponding to the preceding work. Some of the results presented herein are similar to those of Koh [ 2 ] . Here we use the following notations. For our purpose we shall restrict x and y to the first orthant of IRn which we denote by I. Thus I = Ix E IRn , 0 < xv < m v = l r . ,nl. 2 % We shall use the usual euclidean norm, 1x1 = C xvl A function on v=l a subset of IRn shall be denoted by f (x) = f (x1,x2,...,x ) By Cxl " mn m m1 m2 we mean the product x1 x2.. xn. Thus,Cx j = x1 x2 xn where
.
.
.
The notations x m = Cml,m2,...,mn3. xv 5 y U and xv < y, ( v = 1 ,2 ,n) n nonnegative integers in IR i.e. kv k shall Letting (k) = kl+k2+ kn, Dx
,... .
...+
(12.7.1) while (x'Dx)
0
(k)
kn kl Ic2 axlax2.. axn
.
denotes
...
.
2 y and x < y mean respectively The letters k and m shall denote and mv are nonnegative integers. denote
288
Chapter 12
(12.7  2 ) 12.7.1.
The spaces h
and h'
P
1I
Let 11 be a fixed number in (  m , m ) . By hp we mean the infinitely differentiable and complex valued functions $ ( x ) which are defined on I and such that €or each pair of nonnegative integers m and k in IRn (12.7.3) Since $ is infinitely differentiable, the order of differentiation in (xlDx) is immaterial; thus
a
1
1
a
axi) (Xj ax' j =
(Xi
a
1
1
a
(xj ~ ) ( ~ i
for all i,j = 1,2,...,n. The space h is a linear space. Since y P are norms, we have P m,o a separating collection of seminorms i.e. a multinorm. An equivawith lent topology for h may be given by the multinorm {p:l v
k and m traverse a countable index set, h is, in fact, a countaP bly multinormed space. We say that a sequence {$,,I is Cauchy in h P 1I if $, E h for all v and for every m , k , ym,k ($,On) + 0 as v and 1I n + m independently.
As
Lemma 12.7.1. for each k.
If $ ( x )
E
k hP, then Dx$(x) is of rapid descent
Proof. Since 1 a k ixp1/2 . $(Xl,. ..,xi (xi 5 i )
,Xn)
I . .
= x *ixp1/2 . 1
i
ki 1 b.x. j ()I$, a j=O J 1 ax i
'
we have (12.7.4)
(xlDX) [ x 1 P1/2 $ (x) = cx
2k
ICxl2'111
kl
j,=o
...jk1=nO b.[xj] n
where the b . are appropriate constants. 3
x
..+In
4
'1 Jn l.... a xn ax N o w , consider $ E h By J
P
.
Hankel Transform
289
i = l,...,n F i n a l l y , by i n d u c t i o n on k and u s i n g ( 1 2 . 7 . 4 )
w e have
The space h' i s t h e d u a l of h and i s t h e s p a c e of d i s t r i b u ?J P' t i o n s (continuous l i n e a r f u n c t i o n a l s ) on h lJ'
The f o l l o w i n g p r o p e r t i e s are immediate e x t e n s i o n s of t h e one dimensional case.
Using t h e r e l a t i o n ( 1 2 . 7 . 4 )
whenever c a l l e d f o r :
1. I D ( 1 ) , t h e s p a c e of i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t on I , i s a subspace of h f o r every c h o i c e of u. P Thus, t h e r e s t r i c t i o n of any f E h ' t o ID( I ) i s i n I D ' ( 1 ) However 0 D ( 1 ) i s n o t dense i n h
v
2.
.
.
The complex number t h a t f
E
assigns to $
h'
u
by < f , $ > . W e a s s i g n t o h' t h e following topology:
E
h
P
i s denoted
u
a sequence { f , ) converges t o f J
for all $
E
h
.
E
h' i f < f  f j , @ > + 0 a s j + ?J
m
?J
t h e r e exists a p o s i t i v e c o n s t a n t C and a For each f E h ' ?J nonnegative i n t e g e r r such t h a t
I
lJ
max ymIk ($1. O I m ,kz r be a Locally summable f u n c t i o n on I such t h a t f (x) i s and Cxl u+1/2f ( x ) i s a b s o l u t e l y i n t e g r a b l e a s ( xI + v = 1,2,. ,n. Then f ( x ) g e n e r a t e s a r e g u l a r generaf i n h ' d e f i n e d by =
p:
L e t f (x)
of slow growth on 0 < x y < 1, lized function
..
?J
.....( f ( x l I x 2 , ...,xn) $(x1,x2 ,...,xn)dxldx2 .....dx n' m
m
< fI+> = 0
0
4
T h i s s t a t e m e n t f o l l o w s from
E
hV.
t h e mean v a l u e theorem f o r ndimensional
Chapter 1 2
290
i n t e g r a l s (See Fleming CllIp.155) and t h e f a c t t h a t 9 i s of r a p i d descent. Operations on h
12.7.2.
and h 1 1.1
!J
I n t h i s s e c t i o n w e perform some o p e r a t i o n s on h following manner. Lemma 12.7.2.
lJ
and h' i n t h e !J
For any p o s i t i v e o r n e g a t i v e i n t e g e r n and f o r
any u I t h e mapping $ ( x ) + C x l n 9 ( x ) is an isomorphism from h o n t o !J h;l+n* It follows t h a t f ( x ) + I x l n f ( x ) d e f i n e d by . d
Cxlnf ( x ) I 9 ( X I > = < f
i s an isomorphism from h t
u+n
Proof. If 9
E
(XI
I
[XI"$
(XI >
o n t o hl
!J*
hlJI t h e n
I.
= sup cx Im(xlox) kCx 1u1/2n
lJ+n ( [ X I n $ )
'm,k
Cxlncp(x) I
I
 1y 1m I k ( 9 )
W e now d e f i n e t h e following o p e r a t o r s on h
= x !J+1/2 Nt!J
N
M
lJ
i y
i
a xlJ1/2 axi i
= N1!JN21J..
= xu1/2 i
..
,NnU = [ x ] ~ ' + ~ / ~
p+1/2
Q = xi
N  l cp = 2lJ
xl+1/2
axl.
a xu+1/2 axi i
Also, w e d e f i n e an i n v e r s e o p e r a t o r t o N N:;
2
!J1/2
I
v
an
....ax*
 u 1/2
[ j
as f o l l o w s :
9 (t?X2,.
m
,x2tp1/2~(x11t
.., x n ) d t I...~x
n) d t
m
and so on.
Ctl!J1/2Q(t)dtR,.. That
Nil
i s t r u l y the inverse t o N
lJ
., d t n .
f o l l o w s from t h e f a c t t h a t Q i s
I
Hankel Transform
291
infinitely differentiable and of rapid descent. Lemma 12.7.3.
N 0 is an isomorphism from h
Lemma 12.7.4.
M + is a continuous linear mapping of h U+1 11 on to h
Lemma 12.7.5.
M
P
onto h?J+l.
?J
! i *
U
N
P
2p + l
= [x]~'~'*
ax1,.,..
n an axl.. a xn [ 1 ?J1/2 i=l
=
.
is a continuous mapping of h
(2).
into itself. A
A
In the dual spaces, we define N and M as transpose differenu Fi tial operators by (12.7.5)
f Ehk,
(12.7.6)
f
E
$
E
hL+l, +
h?J+l, E
hP.
A l s o , we can define
These definitions are consistent with the usual meaning of transpose differential operators in the sense of Chapter 5. In views of lemmas 12.7.3, 12.7.4 and 12.7.5, we have .L
Lemma 12.7.6. (I) The transpose differential operator N defined ?J by (12.7.5) is a continuous linear mapping of h' into hL+l. ?J
(2) The transpose differential operator Mudefined by (12.7.6) is an isomorphism from h' onto h'. P+1 v r
(3) The transpose differential operator M
h
N given by (12.7.7)
l J ? J
is a continuous linear mapping of h ' into itself. lJ
12.7.3.
TheJiankel transformation in nvariables
The structure of a distribution formulated in Section 12.7.1 enables us in this section to describe the distributional setting of Hankel transformation in nvariables in the following manner.
292
Chapter 12
We define the ndimensional classical Pth order Hankel transformation by
1.. ...I$ (xl,. ..,xn)Z'J, m
=
m
0
0
( 2 ) dxl..
...dxn
$,
the Hankel transform exists where Z = x 1y1+...+ xnyn. For 1.1 2 for every 9 F h This is due to the fact that $ is infinitely P. of rapid descent as 1x1 + ; while Zf J ( Z ) = O ) ' " Z ( differentiable and P as Z + O+ and it remains bounded as IZI + m. These properties of $(xl, xn) also ensure the validity of the classical results (12.1.8) and (12.1.9) when extend to ndimensional and these results are given by
...,
(12.7.9)
U Y , , . ..rYn)
=
q 4 4 X l,*..,x n1
=
1..
a0
0
n
... J0$ Rl,...,tn)i=n 1 (tiYi)% m
and (12.7 .lo)
where y = (y,
,...,yn) and x =
(x,
,...,xn1 .
Note that these formulae are also valid if $ E h 11. and (12.7.10) we can make the following inclusions. 1. M P $ and IH'l4 P which belong to h
are functions of (yl
,...,yn) and
By (12.7.9)
(xl
,...,xn
P*
and
then M u $n 2. If a sequence {$,I + 0 in the sense of h PI + 0 in the sense of h
+ 0
P*
,...,
3 , In (12.7.10) and (12.7.91, $(xl x,) and $(yl are called the antitransforms (or inversetransforms) of ,Y,) and 4 (X ll... #xn 1 4 (y
Also, by (12.7.9)
and (12.7.10)
we have
,...,y,)
)
Hankel Transform
Y Mu @
M;
(XI
= 9
293
.
(XI , x= (xl,.. ,xn) and y= (yl,.. .,Yn1
*lJ
Hence, we may conclude that or My is self reciprocal. Therefore, lJ IH; and My are reciprocals automorphism of each other on h if lJ
9
E
hlJ,then M 6 lJ
E
H
lJ
lJ;
.
The above results permit us to make the following definition: The Hankel transformation of order l~ of a distribution V E h' (x) lJ is the distribution in H ' (y) which is denoted by M V (MyVx whenever lJ lJ lJ we desire to make the variable precise ) and defined by (12.7.11)
<M; VX, 9 (Y,
..tYn)
> =
'Vxi
9 ( ~r 1
ryn)>
V9
E

hlJ
This is a definition by transposition given in Section 5.1 of Chapter 5. To conform with established terminology, we shall say that every generalized function (or distribution) belonging to h' is a M lJ v transformable generalized function (or distribution) in nvariables. Also
,
by (12.7.10)
(12.7.11')
< MYV
lJx
I
M :
(12.7.11) is equivalent to
,...
,X ) > =
@(x,
n
..,xn)bV
@
E
h
?J
.
Since M 9 belongs to h we deduce from (12.7.11') that M I J V belongs lJ lJr to h' and hence we conclude that transformation defined by M in lJ lJ (12.7.11) is an automorphism on h' lJ
.
If the distribution V is associated with a summable function h (xl,.. ,xn) on I, then (12.7.11) leads to the equality m n 4 Ih(tl,...,tn) ( n (t.y.) 1 1 (12.7.12) 0 i=l
.
.....
CO
in accordance with (12.7.9). Thus, we may infer from this distributional setting that the Hankel transformation of generalized functions in nvariables generalizes the Hankel transformation of functions in nvariables. Problem 12.7.1 If 'Sn(a,R)
denotes the Surface distribution in the sense of
294
C h a p t e r 12
S e c t i o n 4.5 o f Chapter 4, t h e n show t h a t
We now e s t a b l i s h some t r a n s f o r m a t i o n f o r m u l a e on h Lemma 12.7.7.
(12.7.13)
1
Let
2
 z'
(N,,0)
(12.7.15)
H$Cxl20 (X)) =
(12.7.16)
M,,
E
E
and h l .
h,,, t h e n
M,,+l (CXI0) = N,, H,, 0 (XI
(12.7.14)
If 0
If 0
P
h,,+l,
= CylM,, 0
(M,, N,,O)
M,, N,, M,,0
= (1)"Ly12m,, 0.
then
(12.7.17)
M,, ( C x l 0 )
(12.7.18)
M,,(M,,*)
=
M,, M ,,+10
= CYIM,,+~$.
P r o o f s . The p r o o f s of t h e s e formulae c a n be seen i n Koh C n l , pp. 432433.
The above lemma e n a b l e s us t o p r o v e t h e f o l l o w i n g theorem whose proof follows a n a l o g o u s arguments t o t h o s e of S e c t i o n 12.5.2 u s i n g t h e a p p r o p r i a t e d e f i n i t i o n of t r a m p o s e d i f f e r e n t i a l o p e r a t o r s (12.7.5) , (12.7.6) and (12.7.7). Theorem 12.7.1.
I.I 2
Let
 i.
I f V E h;,
i+i,,v
(12.7.19)
rnJ (1)" C X l V )
(12.7.20)
m,,+l(i,,v)
= (1)ncyllH ,,v
(12.7.21)
M$(l)"[xI
2V) = M
=
.
L
N,, H,,V A
!J
(12.7.22) If
v
E
h;,
then
(12.7.23)
m,,ICxlV)
(12.7.24)
XI,, (M,,V)
A
= M,,M,,+lV
A
=CYIM,,+~V.
then
Hankel Transform
295
12.8,Variable Flow of Heat in Circular Cylinder In this section we use the preceding theory of generalized Hankel transformation in one variable to solve flow of heat in circular cylinder. By a generalized boundary condition we mean a generalized function in h'V ' which is approached by the solution at some particular boundary. Specially, the problem we wish to solve is the following : Find a temperature function V(r,t) on i(r,t) , r > 0, t < , 0 < 0 < 2t) 1 that satisfies with k thermal conductivity of the diffusion equation: m
<
(12.8.1) and the following boundary conditions: (a) As t + O+, V(r,t) converges in some generalized sense to the distribution f(r). We consider here that r varies from 0 to R where R is the radius of cylinder. Also, we consider V(r,t) and f (r) have bounded support (relative to r) and belonging to E ' ( 1 ) where I = 0 < r < R. The differential equation (12.8.1) can be converted into a form that can be analysed by our zero order Hankel transformation by using the change of variable
Here again u(r,t) and g(r)
e'(1).
E
Accordingly, (12.8.1) becomes
applying M o to (12.8.2) , formally interchanging Mowith setting U ( p ,t) = M o Cu (r,t)1, we can convert (12.8.2) into dU(p,t) dt
+
k
P
2
U(p,t)
The boundary condition suggests that Section 12.4.1 we may write
= 0.
A (p)
=
M o g (r) so that by
and
Chapter 12
296
(12.8.3)
A ( P ) = < g (XI I
& J0(xp)
>
.
Furthermore, Theorems 12.4.1 and 12.4.2 state that, for each fixed t > 0, A ( P ) ekp2tis a smooth function of p in L ( 0 , m ) . Therefore, we may apply the conventional inverse Hankel transformation to get our formal solution: (12.8.4) That (12.8.4)
is true the solution can be shown as follows.

First of all, Jo(rp) and J,(rp) are bounded on 0 < rp < and 2 2 ekp t,  kp for T 5. t < , 0 < p < m . These facts and the Theorem 12.4.1 allow us to interchange the differentiation in (12.8.2) with the integration in (12.8.4) since at every step, the resulting integral converges uniformly on every compact subset of 0 < r < m. 2 Since emkp G p Jo(rp) satisfies the differential equation (12.8.2) for each fixed p, we can conclude that u(r,t) also satisfies (12.8.2). Hence V (r,t) satisfies (12.8.1)
.
We now prove our boundary condition.
A s a function of p,
(12.8.5) is smooth, and for each fixed t > 0, it is a member of L ( 0 , m ) by virtue of Section 12.4.1. Thus, (12.8.5) satisfies the conditions under which the conventional Hankel transformation is a special case of our generalized Hankel transformation. According to (12.8.4) , its Hankel transform is u (r,t) , so that, for any 0 E Ho and # = Mo 4 , our definition of generalized Hankel transformation yields (see equation (12.4.3)),
The integral on the righthand side converges uniformly on OLt< because its integrand is bounded by
l < g(XI, KP J0(xp)
>
@ ( PI I E L ( O , m ) .
Thus, we may interchange the limiting process t integration to get
+
O+ with the
m
lim = j
EPJ0(xp)> Q(p)dp.

Bessel Series Again by (12.4.3)
I
297
the right hand side is equal to
we have shown that, in the sense of convergence in HA,u (r,t) + g (r) In otherwords, V (r,t) + f (r) in a generalized sense i.e. as t + O + . in the sens,eof HA. Problem 12.8.1 Prove that (12.8.4) satisfies the differential equation (12.8.2) for 0 < r < m and 0 < t c a. 12.9. Bessel Series for Generalized Functions F r m now we shall be concerned with the second part of this chapter which shows that the distributions having support on Co,a] can be developed in a Fourier Bessel and Eessel Dini series which converge to the distributions on the space of conventional functions contained in a particular space I D ( 1 ' ) where I' denotes the interval [O,al. The results presented in this part bear the close resemblance as indicated in Lavoine [ E l . 12.9.1.
Statement
The construction of the results presented in this section depends upon the properties of the Bessel function given in Watson c11.
i.
Throughout this section we take v The Bessel function of order v is defined by (see Problem 1.4.1 of Chapter 1)
and A j = 1,2,3 denote the positive roots of J (ax) = 0 in increaj' V sing order. We put f
elsewhere. For each v,J1 (A.x) forms an orthogonal system: v 7 a I x Jv(A.x)Jv(Akx)dx = (12.9.1) 7 0 . ) a j :+ ; J
[
I
k = j.
By T we denote a distribution with support contained in 1'.
(This
298
Chapter 12
means that < T I $ > = 0 for each function +(XI whose support is exterior to I!.) By every distribution with bounded support (see Section 4.1.3 of Chapter 4), T is of finite order, and this order of T we denote by m, m = 0,1,2,...
.
We associate the numbers A . T with T as J (12.9.2) and the sum (12.9.3)
Tn =
n
1 Aj
j=1
(T) J,(X.x) v 3
which is evidently a distribution with support in 1'. It is important to remark that if v is neither an integer nor 0, then in order that A . ( T ) exists in general, it is necessary to be 3 v > m1. This is because the derivatives of xJv(A x) of order m are j not continuous at the origin. In (12.9.2), the A . (T) are called the coefficients of the 3 FourierBessel series of T. If T = f(x) a summable function with support contained in I', then (12.9.2) gives a A . (f) = 2 1 f (x)xJv(X.x)dx, l ' a J;+~(A~~ o) 3 and we find again the coefficients of the FourierBessel series in the sense of functions. Also, we say that m
is the FourierBessel series of the distribution T . Now we want to show that it is convergent and equal to T on a certain space of functions. 12.10. The Space B
m, v
We construct in this section the space B in the following m,v manner on which the distributional setting of Fourier Bessel series will be formulated in the subsequent section. Let m be a nonnegative integer, and let x denote a real variable. By Bm,v we denote the space of functions which are m times continuously differentiable on a compact neighbourhood of I' and
Bessel Series
299
which admit on I' a representation of the type 0
(12.10.1)
with the conditions that either
v >
m  1 or
v
= 0,13,...,
and in all
m
1 la. ($1 I A 3m is convergent. Here a. (0) are j=1 3 3 the numbers which do not depend upon the variable x but depend upon the choice of the functicm Q (x) in B,,,. these cases the series
For x = 0 we have x Jv(A.x) = 0. 3
Q (0) = 0 when Q
(i)
Bm
E
A l s o , by the property of A
J (A.a) = 0. v
3
(ii)
IV
Hence, we have by (12.10.1)
for all integers m 2 0.
given in Section 12.9.1, we have Therefore, we obtain by (12.10.1),
Q (a) = 0 where Q
E
j
Bm
,V*
Hence, we conclude that (i) and (ii) are the necessary conditions for (0 E B and in particular for Q belongs to Bo,v. Let us now m,v further state the property of Bm ,V*
Consider a function Q [x) of Bm,v.
If there exists an interval
1" containing I' , on which Q (x) is .m times continuously differentia
ble. A l s o , if a(x) be an infinitely differentiable function whose support is a neighbourhood of I" and such that a(x) = 1 on 1'. Then a (x)0 (x) is m times continuously differentiable with bounded support and as a consequence belongs to the space IDm of Section 2.3 of Chapter 2. Hence
 SP'
then we Will say
S
P
+
Q(x) >
t
0 for each Q
, or m,v
T on B
of B,,,, that lim S = T on Bm,v. P P+
Chapter 12
300
12.11. Representation of a Distribution by its Fourier Bessel Series The results of Section 12.9 and the construction of the space B given in the preceding section enable us in this section to m,v formulate the distributional setting of Fourier Bessel series in terms of the representation of a distribution on Bm by its Fourier IV Bessel series which we outline in the following manner. Theorem 12.11.1. Let T be a distribution of order m with support contained in 1'. Then we have lim nOn
Tn = T
Bm,vI o r , more explicity in the sense of series ce
T
=
1 AJ. (T) Ja(A.x) V I
j=1
Before giving the proof of this theorem we will need the following three lemmas which enable us to formulate the proof of this theorem. Lemma 12.11.1.
Let 4 belong to B and if we put m,v m
Then 1. the first m derivatives of every neighbourhood of 1';
2. we have on I' that $:h)
(x)
I$

1
(x) exist and are continuous on
4 (h) (x) = 0 where h = 1,2,... ,m.
Proof. By the known properties of Bessel function (see Watson [l]) and considering certain conditions on a,($) (see Section 12.10) J m .h we can show easily the uniform convergence of 1 a j x Jv (A .x) j=1 dxm I on every bounded interval of 1'. Hence, l., is established. By the description of B given by (12.10.1) , we have d;, (x) = $ (x) and m,v consequently 2 . , is also established.
($)a
Lemma 12.11.2.
Let m
d;,(x)
=
1 a. ($1 J
j=n
x J,,(X~X) , Y 4
E
B ~ , ~ .
Bessel Series Then
0, (x)
+
0 as n
+

and
$dh)
(x)
301
...,m.
0 uniformly for h = 1,2 ,
+
proof.^ It is easy to show that, for h positive constant
=
1,2,...,m
with K is any
which tends to zero by virtue of the properties of BmIV. If T of order m has support contained in I', then
Lemma 12.11.3. we have
<TI$(XI > =
(XI >,
v
Bm,v*
Q
This is a consequence of Lemma 12.11.1 and of (Schwartz C11, Chapter 111.7, Theorem XXVIII). Proof of Theorem 12.11.1. (12.11.1) By
Lemma 12.11.3,
(12.11.2)
+
It is sufficient to show that 0, Cp
B
m,v
we can write
NOW, by (12.9.3)
E
,
(12.9.1)
4, (X)>
and (12.9.2)
n
m

, we
4,
(X)>.
have
a
n
Hence, by (12.11.21,
which tends to 0 as n + m . Let a(x) be an infinitely differentiable function whose support is a compact neighbourhood of I' such that a(X) = 1 on 1'. Then <TIJn(x)> = q,a(x)J,(x)> Now, by the.Lemma 12.11.2, a(x) $,(XI 0 in the space of f l o f Section 2.3 of Chapter 2 and consequently T belongs to the topological dual IDfm of IDrn,
.
+
302
Chapter 12
hence
CTr a
(x)qn (x)>
+
0 as n
+
.
Examples. 1. If 0 5 c 2 a, the Dirac measure 6(xc) which is such that < 6 (xc),$ (x)> = $ (c) is a distribution of order zero having support contained in 1'. For v 2 the Theorem 12.11.1 yields the representation J (cX.) (12.11.3) S(xc) = 1 J;(Ijx) a j=l Jv+l(aAj)
%
00
defined on B O I v . Moreover, it is easy to verify this equality by means of Lemma 12.11.3 and formulae of Section 12.9. Note that (12.11.3) gives 6 (x) = 0 and 6 (xa) = 0 which do not yield a contradiction because if $ belongs to B0 , v ' then we have <6 (x),$ (x)> = 0 and <6 (xa)I $ ( x ) > = 0. 2. Let the function g (x) be defined by xv(1x2)a,
0
<
x
c
1
g (X) = I
$
elsewhere,
(0
where v > and a is not an integer (a > 1). Also, g (x) has not any Fourier Bessel expansion in the sense of functions. But in the sense of distributions, Fp g(x) (which is of the order a' = integer part of a) according to the Theorem 12.11.1, is represented on Bi, by the series (2) I V
Note that v must either be a nonnegative integer or satisfy v > a'1.
12.12. Other Properties of the FourierBessel Series The results of the preceding sections enable us to formulate a uniqueness theorem of Fourier Bessel series in a distributional setting which we term as other properties of the FourierBessel series. To prepare for this section we first need the following result. Theorem 12.12.1. The representation of T on B by a FouriermIv Bessel series of order v is unique. In other words, if
Bessel Series
on Bm
IV'
303
then Aj = Aj (T).
Proof. x J v (Akx) belongs to B for each k 2 1. Now by m, v Theorem 12.11.1, we have
it follows that Ak = Ak(T).
Hence, by (12.9.1),
Therefore, Aj=Aj (T).
Magnitude of the coefficients. We give now the following result by means of which one can conclude the magnitude of the coefficients. Theorem 12.12.2.
We have
where K is any positive constant. Proof.According to Section 5.4.3 of Chapter 5, T is equal on an arbitrary neighbourhood of I' to the (m+l)th distributional derivative of a measurable function f(x) which is bounded on this neighbourhood. It follows that A.(T) can be written in the form 3
Now by differentiation and known properties o€ Bessel functions (see Watson C11) we can obtain the desired result of this theorem. Properties of a given series. We now give below the following result which governs the properties of a given series. Theorem 12.12.3. Let A j = 1,2,..., be a set of numbers with j f 1 < HAm+', for a large enough H any positive constant such that [A. m 7 j (in the number j . Then the series 1 A JA (A . x ) is convergent on Bm j=l j v 3 fV sense of (12.10.2) )
.
Proof. For each 4 in Bmfv,Lemma 12.11.3 m
ca
gives
304
Chapter 12
and by (12.9.1) we can show that the modulus of this given series is majorized by the convergent series
where K is any conventional number. 12.13.
The Subspace Bm fo
B mlv
As remarked in the Section 12.10, we construct in this section and present some result of distributional the subspace Bm of Bm IV setting of Fourier Eiessel series on Bm. By % we denote the space of functions +(x) which are m+l times continuously differentiable on an interval containing I' and such (x) is on 1' and that 9
The importance of the space Bm can be seen because it contains the space IDm+2 (I1) of functions which are (m+2) times continuously differentiable with support contained in I' and hence also contains the space ID (1') of infinitely differentiable functions having support in 1'. Theorem 12.13.1.
Bm is a subspace of Bm,".
This result can be obtained from the following lemmas. Lemma 12.13.1. (12.13.1)
Let the FourierBessel coefficient of $(x)/x be
cj ($/XI =
a
2 a2J:+1
I
(ahj) 0
9 (XI J v (Ajx)dx.
If 9 belongs to Bml we have (12.13.2) where M is any conventional number. Lemma 12.13.2.
If 4 belongs to Bml then we have on I', m
The proofs of these lemmas need many calculations and we give the
Bessel Series outlines of them. (12.13.1) 2 a ' 2 Jzl(aX )A ( v , @ ) where j j
305
can be written c. ($/x)= 3
a
First we take
in Bo.
$
Then we have
where
and
By the structure Bol we have for 0 5 x
IO"(x)I < H =>lO'(x)I
(i)
< a < H x and 16(x)I <%x2
where H is a conventional number. Also, when 0 5 x 2 X i 6 with large enough j I we have IJv (Xjx) I < H X\l xv where HI is another conven1
3
zvt;6 Therefore, with 6 = 2b+5
tional number.
, we
obtain
.6
I where M1 = 2v+6HH1.
A
(ii)
j
Hence, we finally get
< M
1
X5/2 j
By the expansion of Jv (z) (see Watson [ll)
where 8 = v
$ + $ and
, we
have
bv (z) is a bounded function for large enough
z > 0. Now, by putting the value of J v(z) with z = h . x from (iii) 3 to J v (X.x) in A', we obtain 1 j 2 1/2 4v21 (iv) A; < CI1,I + 8 1121 + II,l] where a
J cos ( X x8)
I1 = J
.6
j
$J IX)
x1'2dx,
Chapter 12
306
I2 = A;
7
3'2
.
6
sin (A.xB) 7
Q (x)
x~/~~x,
a I3 = A 5/2 bv (hjx)4 (x) x  ~ / ~ ~ x . j .6 *j
Further, if we integrate I1 by parts two times by using (i) and property $ (a) = 0 together with the fact Xi6 + 0 as j + 0 , then we J can conclude that there exists a certain number G1 such that for j > lo. Also, integrating by parts one time to 12, we obtain for j > jo
(vi) where
G2
is a certain number.
If j > j, then there exists a number K such that bv ( A .x) < K for U 3 A T 6 5 x 5 a and by (i)I we have 7 a
J
and consequently there may exist a number G3 such that (vii)
lfgi
Let us denote M2 =
< G ~ hj 5 / 2
for j > jo.
4v2 1 CG1 + g
2 4
(3
G2
+ G31 I
then by making use of (iv), (v), (vi) and (vii) , we obtain (viii)
A!
I
M
2
Pinally, since A. (v,Q) 3
IAj (v,Q)
A  ~ / ~ for j > jo.
j
<
A.+A! I we get by combining (viii) and (ii)
I
<
1
7
(M1+M2) h 5/2 f o r j > jo. j
Take now Q in Bm. If we integrate A.(v,$)mtbes by parts by 7 using the properties of Bessel function (see Watson [l]), then we obtain m
A. (v,4) = (1) 7
x  ~A. (v+m,f) j
i
Hence by (12.13.41 I IAj ( v I Q ) 1 <(M1+M2) Ajm5/2 (which where f E B 0' is also true for v+m). Finally, we deduce (12.13.2) because Jv+l (axj)
307
Bessel S e r i e s
i s of o r d e r
,I% as
j
j
f
The Lemma 12.13.2
. i s a consequence of t h e Lemma 12.13.1 and of
t h e f o l l o w i n g r e s u l t from t h e t h e o r y of F o u r i e r  E e s s e l series of f u n c t i o n s (see Watson Cll, S e c t i o n s 18.24 and 18.26):
uniformly on Ca,al,a
I f m = 292,
Remark. M,ITZS+1/2s 1
> 0 , a s n +
s
i n Theorem 1 of
m.
g i v e s Ic. ($/x) I < 3 (Tolstov Cll, S e c t i o n 8.20) , which
2 1, then (12.13.2)
imposes more r e s t r i c t i v e c o n d i t i o n s on $ (x) /x t h a n our c o n d i t i o n s . (See a l s o Khoti Ell.) C o r o l l a r y of t h e Theorem 12.13.1.
I n t h e Theorem 1 2 . 1 1 . 1
and
by Bm w i t h t h e c o n d i t i o n t h a t e i t h e r w e can r e p l a c e Bm IV i (ii)m = 0 and v 2 2 or (iii)m = 1 , 2
12.12.3
(i) v = 0 , 1 , 2 , and v > m1.
...,
,...
 L,
12.14. EesselDini S e r i e s
I n t h e above s e c t i o n s , w e have shown t h a t t h e d i s t r i b u t i o n s wikh s u p p o r t i n C0,aI a r e r e p r e s e p t a b l e by a FourierBessel series on t h e space B where w e impose t h e c o n d i t i o n t h a t i t s f u n c t i o n s m,v This c o n d i t i o n i s r e p l a c e d by a n o t h e r c o n d i t i o n v a n i s h a t x = a. f o r c e r t a i n problems when w e s t u d y BesselDini s e r i e s a s g i v e n below. T h i s c o n s t r u c t i o n and r e s u l t s of t h e p r e s e n t s e c t i o n are p a r a l l e l t o t h a t of S e c t i o n 12.9. Statement
12.14.1.
We take v
2

i,
H is a real parameter; I ' d e n o t e s t h e i n t e r v a l
CO,a]; x i s a r e a l v a r i a b l e ; and B;,
j = 1,2,..., are t h e p o s i t i v e We p u t r o o t s of t h e e q u a t i o n axJ:(ax)HJv(ax) = 0. J
if v > H I i f v = H
1
Iv(Box),if v
Ry I v w e denote t h e modified Bessel f u n c t i o n , I
and
$ ,
i s t h e p o s i t i v e number such t h a t + i B o
equation
V
(x) = i'JV (ix) ,
i s a r o o t of t h e
Chapter 12
308
z'CazJ:
[az)

HJ\,(az)I = 0,
so that
For each V , G ~@,XI and the J~ ( 8 .XI 3 system on 1' such that
j
1,2,.
=
.., form an orthogonal
where ,if v
>
H,
, i f v = H,
n0=
we put G;
@,XI
=
elsewhere and (12.14.2)
J A (6.X) = v
3
elsewhere.
To a distribution T with support contained in I' and of order m (see Sections 4.1.3 and 5.4.3 of Chapters 4 and 5) we associate the numbers $(T)
(12.14.3) and the
sum
=
B. (T) =
I
1,< T, xGv (H,x) > 00
I a
Bessel Series
309
which evidently is a distribution having support 1'. It is important to remark that if v is .not an integer nor 0, then in order that 8.(T) exists in general, it is necessary to be 7 v > m1. This is because the mth derivatives of xJv ( 8 .x) are not 1 continuous at the origin. In (12.14.3), the B. (T) are called the EesselDini coefficients 7 of T. If T = f(x), a summable function having support in I', then we obtain the B. (f) to be the BesselDini Coefficients in the sense 3 of ordinary functions. We further say that m
is the BesselDini series of the distribution T. NOW our purpose is to show that it converges and is equal to T on a certain space of functions, 12.15. The Space on which the In this section we construct the space % rmrv distributional setting of FourierDini series is to be formulated in the subsequent section. Let m be a nonnegative integer and x a real variable. By we denote the space of functions $(x) which are m times %,m,v continuously differentiable on a compact neighbourhood of I' and which admit on I' a representation of the form m
$
(XI
=
do ($1 xGV (H,x) +
1 d. (0)xJv (Bjx) j=1 J
with the conditions that either v > m1 or v = 0,1,2,...,
and in all
m
1 Id. (0) lf3m3 is convergent. Here d.3 (0) are j=1 3 the numbers which depend upon the choice of $(XI in rmrv.
these cases the series
We give below (Theoremsl2.6.5 and 12.6.6) definitions of the independent of Bessel functions. subspaces of ,m,v
a
The existence of
,Y
0
E
%,m,v
can be shown as that
EkIm
Two distributions T and
S
having support in I' and of order z m
Chapter 1 2
310
Convergence on
a
I
f
Let S I p E IN, be a sequence of distributions with support P contained in I' and of order a. If there exists a distribution T having support on I' of order a such that
+spI
(12.15.1)
then we will say S
Q (XI> = 0, Y Q
E
%,m,v,
as p
t 0 0 ,
or that lim S = T on %tmtv P+" P 12.16. Representation of a Distribution by its &SEelDlni +
P
T on
Series
In this section we obtain results for the distributional setting of BesselDini series which would be similar to those of FourierBessel series given in the Sections 12.11 to 1 2 . 1 2 . Theorem 1 2 . 1 6 . 1 . Let T be a distribution of order m having support in 1'. Then we have

lim Tn
n
=
T on %,m,v
i
or more explicity in the terms of series m
+j=1 1 B.1 ( T ) J ; ( B 3. x ) on %,m,v*
T = B~(T) G;(H,x)
Theorem 1 2 . 1 6 . 2 . The representation of T on by a rmr V EesselDini series of order v and parameter H is unique, i.e. m
T = B~ G;IH,X) +
then B
j
= Bj
B . J ~ ( B . , X )on j=1 I v 3
if
% rmrv
(T) , where j 2 0.
Magnitude of the coefficients. We now give the following result by which one can conclude the magnitude of the coefficients. Theorem 1 2 . 1 6 . 3 . positive constant.
We have IB. (T) I 3
<
K B';+3'2
where K is any
Properties of a given series. The preceding results enable us to make the following result. Theorem 12.16.4. Let B j = 0,1,2,..., be a set of numbers j r with M any positive constant such that IB. 1 < MBY+l for j being a 3
Bessel S e r i e s
311
m
Then t h e series
l a r g e enough number. On
( i n t h e sense of
%,m,v
12.16.1.
The subspace Bm
(12.25.1) )
of
.
1
B . J;(B.x) j=1 7 J
i s convergent
%,m,v
m i s t h e s p a c e a l r e a d y d e s c r i b e d i n S e c t i o n 12.13.
B
Theorem 12.16.5. Corollary.
B
m
i s a subspace of
I n t h e Theorems 1 2 . 1 6 . 1
%, m , V .
and 1 2 . 1 6 . 4 ,
one can
by Bm w i t h t h e c o n d i t i o n t h a t e i t h e r v = 0 , 1 , 2 , . . . , 1 e i t h e r m = 0 and v T, o r e i t h e r m = 1 , 2 , . . . . , and v > m1. replace

Note.
The proof of Theorem 12.16.5
i s a n a l o g o u s t o t h a t of
Theorem 12.13.1 o f S e c t i o n 12.13 by r e p l a c i n g t h e r e f e r e n c e s
(Watson
Cll, S e c t i o n s 18.24 and 18.26 by S e c t i o n s 18.33 and 1 8 . 3 5 ) . 12.16.2,Another
s u b s p a c e of
Theorem 12.16.6.
By
%m,v
%,O,V
w e d e n o t e t h e s p a c e of f u n c t i o n $(x)
which are t w o t i m e s d i f f e r e n t i a b l e on a n i n t e r v a l c o n t a i n i n g I ' and such t h a t Q" (x) /x i s bounded on I '
0' (a)  (H+L) Q (a)
,
$ (0)
=
$ I
(0) = 0 , and
= 0.
P r o o f . The proof i s a n a l o g o u s t o t h a t of Theorem 12.13.1
w e p u t f (x) =
and u t i l i z e t h e r e s u l t s of
( T o l s t o v [l],
if
Sections
8.22 and 8.23). 1 2 . 1 7 . An A p p l i c a t i o n of t h e BesselDini
Series
T h i s s e c t i o n p r o v i d e s an a c c o u n t of t h e u s e of t h e p r e c e d i n g t h e o r y of B e s s e l D i n i series t o f i n d o u t t h e s o l u t i o n of t h e problems of h e a t flow i n a c y l i n d e r ' o f i n f i n i t e l e n g t h . S p e c i a l l y , t h e problem w e wish t o solve i s t h e f o l l o w i n g : Find a t e m p e r a t u r e f ( r , t ) (which i s a f u n c t i o n of r and t) such t h a t 1. it i s d e f i n e d f o r r i n I ' = [O,al; 2.
it s a t i s f i e s t h e p a r t i a l d i f f e r e n t i a l equation
f o r 0 2 r 2 a , where v 2 0 , w i t h g (t) b e i n g an i n t e g r a b l e f u n c t i o n €or t > 0 and S (r) i s a d i s t r i b u t i o n w i t h s u p p o r t i n 1';
Chapter 12
312
3. it satisfies the conditions
(12.17.3)
f (r,t) is bounded when r is bounded
I
a Hf (a,t) = 0, H < u. aar f (r,t) r=a To solve this problem, we identify f(r,t) with a distribution in r on l D ( 1 ' ) that eliminates the condition 1 and the restriction 0 < r < a in (12.17.1). (12.17.4)

By Theorem 12.16.1 and the corollary of Theorem 12.16.5, distribution S (r) can be represented by
the
(12.17.5) where J; (6 .r) is given by (12.14.2) and 1
S .=
'
5
<S
(r), r J u (Bjr)>.
j
The structure of S(r) given above enables us to consider the representation of f(r,t) such that m
with F . (0) = 0. Now, we verify that (12.17.6) satisfies the 3 conditions (12.17.2) , (12.17.3) and (12.17.4)
.
Since we take F . (0) = 0 and hence f (r,t) given by (12.17.6) 3 satisfies the condition (12.17.2). From Watson [l], we know that all the J; (8.r) is bounded for 1 u 2 0 (see also Theorem 12.16.4) and hence fw,t) represented by (12.17.6) satisfies the condition (12.17.3)
.
By
the property of 8 . given in Section 12.14.1, we have
for r = a. (12.17.4).
3
Hence, f (r,t) given by (12.17.6) satisfies the condition
NOW, by putting the values of f (r,t) and and (12.17.5) in (12.17.1), w e obtain
S
(r) from (12.17.6)
Bessel Series
313
But, we have by the recurrence formulae of Bessel function (see Watson [ 11) that
Thus, we obtain
Consequently, by making use of (12.17.8)
,
(12.17.7) takes the form
m
From this result we observe that (12.17.1) can be verified in a distributional setting if we solve the equation 2
B 7. F7. (t) +
k&
F j (t) =
s 7.g(t)
with F. (0) = 0 (which can be solved easily). Hence, we finally 7 obtain 2 2 m t B .ku dule@ 'k J;(B .r) f(r,t) = k 1 S . C l g(u)e I j=1 1 o on
(1').
If we put v 2 0 in (12.17.1), then the present case is a problem of heat conduction in a cylinder of infinite length and of radius a, cooled over its cylinderical surface and heated by spring of intensity g (t)S (r) when f (r,t) denotes the temperature at time t at the point whose distance from the axis is r. The use of Bessel Dini series is already known in the situation of this case when S(r)
Chapter 12
314
is a function (see Carslaw and Jaeger
[2]).
(See
also Section 12.8.)
A theoretically interesting case arises when S(r)= 6(rc)/2nc, c < a. In this case we recall that r is equal to (x2+y 2) % and hence we have
0
<
6 (rc)
2nc
, cp (x,y) >
=
I
1 2n 4 (c cos e , c sin 0
e)ae.
12.18, Bibliography
In addition to the works given in the text we should like to mention some references, which deal with the material of the present chapter. Dubey and Pandey Cll, Fenyo Ell, GUY [ l l r W e e [I], Lions [l], Srivastav [ll, Trione C11 and Zemanian [ 2 1 . Footnotes (1) Zemanian [ S ] has taken k to be a positive integer, we take k
C21,
0.
(2) for the calculation of the coefficients (see Erdelyi (Ed.) v01.2, p. 26).
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..
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This Page Intentionally Left Blank
INDEX OF SYMBOLS Ac,
12
F x , 91
BH,O,v' 311 B 309 H,rn,V' Bm, 304, 311 BIll,V' 298 C($')r
70
hv, 288 hl, 288
My', 292 Hu , 272 : , 272 H M v l270
ID, 19 ID' I 27
80 80 36
Mi',
80 80 Do (b)t 80 (b) 80 I D ( 1 ) ,232, 273 I D ' ( 1 ) , 273 ID(IR"), 23 ID'( W"), 27 ID; (W"), 43 IDo (01,
IDA (01,
IDb+(lR"), IDk,
91
M p , 292
c", 1
ID, ID;, ID:,
,I;..
270 JA , 221 JA(r), 221 JI', 209 JI ' ( r l , 207 L(O,m), 269 IL, 109
43
20
I D V k l 27
m k ( m n ) , 23 ID' k ( l R " ) , 27 ID'l , 89 (ID') I, 89 227, 230 Ea,lll' 227, 232 E:,ul E 231 E 233 Pr9' E(r) , 207 E , 21 E' i 27 f i ( I R n ) , 23 E ' ( I R n ) , 27 FP, 8 \
329
This Page Intentionally Left Blank
AUTHOR INDEX Albertoni, S., 75, 315 Antosik, P., 85, 315 Apostol, T.M., 229, 315 Arsac, Jacques, 105, 315
FOX, C., 227, 268, 318 Friedman, A., 25, 318 Friedmann, B, 205, 319 Fung, Kang, 268, 319
Benedetto, J., 144, 209, 315 Berg, L., 315 Bochner, S., 315 Bredimas, A., 160, 203, 315 Bremermann, H.J., 105, 240, 316, 325 Bremmer, H., 144, 326
Garnir, H.G., 25, 72, 144, 231, 233, 319 Gelfand, I.M. 16, 21, 91, 105, 319 Geradi, F.R, 227, 319 Ghosh, P.K., 144, 319 Giittinger, W., 21, 75, 78, 319, 325 Guy, D.L., 314, 319
Campos Ferreira, J., 85, 316 Carmichael, Richard, D., 225, 316 Carslaw, H.S., 182, 203, 314, 316 Chandrasekharan, K., 315 Choquet, Bruhat, Y., 78, 105, 316 Churchill, R.V., 144, 316 Colombo, S., 144, 157, 182, 238, 244, 259, 262, 271, 317 Constantinesco, 89, 317 Courant, R., 205, 317 Cristescu, R., 105, 317 Cugiani, M., 75, 315 de Jager, E.M., 105, 317, 325 Di Pasquantonio, F., 17, 317 Ditkin, V.A., 144, 317 Doetsch, G., 144, 190, 317 Dubey, L.S., 314, 318 Durand, L., 105, 316 Ehrenpreis, L., 105, 318 Elliott. D., 224, 326 Emde, F., 14, 135, 273, 320 Erdelyi, A., 144, 225, 318 Erdelyi, A,(Ed.), 117, 122, 125, 126, 144, 157, 165, 181, 189, 193, 197, 201, 215, 238, 270, 277, 314, 318 Fenyo, I., 314, 318 Fisher, B., 75, 318 290 , 318 Fleming, W.H., 331
Hadamard, J., 7, 320 Handelsman,Richard,A.,227,320 Hayashi, Elmer K.,225, 316 Hilbert,D, 205, 317 Humbert,P., 194, 322 Ince,E.L.,
320
Jaeger,J.C.,182, 203,314,316 Jahnke,E.,14, 135, 273, 320 Jeanquartier,P.,268, 320 Jones, D.S. 75, 144, 320 Kaufmann,H., 324 Khoti,B.P., 307, 320 Koh, E.L.,275, 287, 294, 320 Korevaar, J., 144, 320 Krabbe, G., 144, 321 Kree, P I 314, 321 Laughlin, T.A., 238, 321 Lavoie, J.L. 200, 321 Lavoine,Jean, 16, 17, 59, 84, 85, 89, 105, 130, 131, 132, 133, 144, 157, 190, 191, 192, 193, 194, 196, 201, 203, 209, 216, 219, 237, 238, 253, 262, 297, 317 , 321 Lew, John,S., 163, 227,320, 322 Loins,J.L. 314, 322
Author Index
332
Liouville, J., 199, 204, 322 Livennan,T.P.G., 144, 322, 325 Lojasiewicz, S., 82, 83, 84, 89, 322 LOSChl F a , 14, 135, 273, 320 Maclachlan,N.W.,190, 194, 322 Magnus, W., 197, 280, 323 Marinescu, G., 105, 317 McClure,J.P., 224, 323 Mikusinski,J,,25, 42, 85, 144,315, 323 Milton, E.O., 105, 132, 225, 316, 323 Misra,O.P., 29, 84, 85, 89, 209, 216 219, 225, 237, 253, 321, 323 Munster, M., 144, 319
Oberhettinger,F., 197, 280, 323 Oldham, K.B.,198, 323 Osler,T.J., 200, 321 Paley,R., 97, 324 Pandey,J.N., 225, 314, 318, 324 Prudinkov,A.P., 144, 317 Roach, G.F., 205, 324 Roas, B., 198, 324 ROOS, B.W.,
324
Roberts, G.E. 324 Rota, G.C., 324 Rudin,W. , 105 I 324 Sato, 324 Sauer,R., 3171 324 Schmets,J., 25, 231, 233, 319 Schwartz,L., 7, 22, 25, 6 4 , 66, 71, 72, 75, 78, 81, 89, 91, 97, 102 105, 108, 141, 149, 205, 324 Shilov,G.E.,16, 21, 91, 105, 319 Sikorski,R., 85, 315, 323 Silva e Sebastia, 25, 84, 85, 87, 105, 144, 225, 324, 325 Smith, M.,
325
Sneddon,I.N., 238, 259, 260, 282, 325 Soboleff, S.L., 325 Soni,R.P., 197, 280, 323 Spanier,J., 198, 323
Srivastav,R.P. 314, 325 Stakgold,I. ,205, 325 SZabO, I., 317, 324 TOlStOV,G.P. 307,3111 326 Tremblay,R.,200, 321 Treves,F., 48, 66197, 102,326 Trione,S.E., 314, 326 Tuan,P.D. 224, 326 Van Der Pol. 144, 326 Vladirnirov,V.S.,64, 326 VoKhacKhoan ,105.,150,205,326 Watson,G.N.,158, 297, 305, 306, 307, 311,312,313,326 Wiener,W., 97, 324 Widder,D.V., 207, 326 Wilde, M.De., 25, 231,233,319 Wong,R., 224, 323
Yosida,K.,. 163, 205, 326 Zemanian,A.H.,
22, 25,38,85,
1 0 5 1 144,225,227,231,233,
235, 244,268,272,273,274, 282,284,287,314,320,325, 326, 327 Zygmund,A., 2041 327.