ANALYTIC REPRESENTATION OF THE DISTRIBUTIONAL FINITE HANKEL TRANSFORM

Various representations of finite Hankel transforms of generalized functions are obtained. One of the representations is shown to be the limit of a certain family of regular generalized functions and this limit is interpreted as a process of truncation for the generalized functions (distributions). An inversion theorem for the gereralized finite Hankel transform is established (in the distributional sense) which gives a Fourier-Bessel series representation of 
generalized functions.


INTRODUCTION.
Zemanan [1] extended Hankel transformations to the distribution space H'.H' is the dual of the space of testing functions H defined as follows" for each real number u, let H { (0,-) ( m is smooth on (0,-) and satisfies (1.1) y ,.u (@) sup Ix m (x -I D) k [x --} (x)] < for each m k 0 2 (I 1) -m,...._ o<x<o= Hu consists of certain distrbutions of slow growth.Then later [2] he obtained a more general result by removing the restriction on the slow growth of the distributions.He defined the Hankel transformation of a distribution of rapid growth in the space B'.p B' is the dual of B, the strict inductive limit of the testing function spaces Bp, b (defined in section 2) as b tends to infinity through a monotonically increasing sequence of positive numbers.
We take advantage of the fact that functions in Bp, b are identically zero after b, to define the finite Hankel transformation for the generalized functions in its dual B' This is done by generalizing Parseval's equation We find that for ,b >the finite Hankel transform / maps B' isomorphically onto the p u,b generalized function space Y' (defined in section 3) The aim of the present p=per is to obtain various representations of the generalized functions in Y' and u,b to find an invesion formula for the generalized finite Hanke] transform which also gives another representation of the members of B' h,b as a Fourier-Bessel series.
Here deotes the open interval (0,).The letters x,y,t and w are used as real variab1s on I.The k th derivative of an ordinary or generalized function f(x) is usually denoted by D k f(x) (though the symbol D k f(x) is also used) X denotes the space of smooth functions that have compact support on I.The topolog9 of D(!) is that which makes its dual the space D'(1) of Schwartz's distributiu,s [3, vol.I, p.65].Let b > 0 be a fixed arbitrary real number.Then for u c R, where R is he nez of re] numbers, we define B { z(x) is smooth (x) : 0 for x b and satisfies ( I)} sup !(x -I D) k Ix -u-; q(x)]l , for each k 0,1,2 (2.1)   o<x<.

B
is a linear space to hich we assign the topology generated by the countable set ,b of seminorms Yk" [u,b is a sequentially complete countably multi-normed space [2].
Classically, for u + O, the finite Harkel transform of a testing function in B,b is defined as @(: where as usual J der.otes the Bes3el function of the first kind of order h and n (n 1,2,3 are positive roots of J (bz)= 0 (arranged in ascending order of magnitude).However @, can be extended to the analytic function of the complex variable z yiw by (z) f (x) j (xzdx. (2.2) Note that (z) is ar analytic function on the finite z-plane except for a branch point at z 0 [4, p. 145 . .Henceforth, the finite Hankel transform of a testing function in B,b shall be defined as the na]ytic function @(z) given in (2.2) and denoted by / The topology of Y is the one generated by using the ),k u,b k 0,1,2, as seminorms.Y is a sequentially complete countably normed space.For further properties el these spaces one can look into Zemanian [4, 2].
For a given testing function @ in Y (2.4) where Ak is cc.nstant.Also (x/n)} [Ju(XAn)/Jh+1(bAn)] is smooth and bounded on 0 x , for -.Consequently, the Fourier-Bessel series (2.5)  Since (>n is of rapid descent as x n and (XXn)--kJ +k x Xn s smooth and bounded on (0,=) for _> -, it follows that the right-hand side of (2.6) converges absolutely and uniformly fcr all x 0 and for every k 0,1,2 Hence the left-hand side is continuous and bounded on 0 --], Moreover, (x ID)k(x -1/2(x)) x h 1/2Lake where A is a constant.
So we see that y< ( ) Also , (x) is smooth on (0,).Hence }, B ,b" It is easy to see that im+ (x)(x) (x) for any  1), then Zn (Tn) iS in general an infinite sum and it need not be convergent.
Note that (i) B' contains every regular distribution that corresponds to a function ,b which is Lebesgue integrable on 0 x < b.In this case we have (iii) Similarly, every regular distribution F, which can be defined by a locally integrable function F(y) through the equation < F, > f" F(y)(y) dy for every in Y belo,gs to Y' Note that F(y) need not be ntegrable ,b' ,b" over 0 y , though typically it would be of slow growth, i.e., for some integer N O, y-N F(y) 0 as y .The above equation also defines the nverse Example i.The fnite Hankel transfo of the delta function (x-k) is given by the equation (4.1)" h a(x-k), (z) < (x-k), (x) >, for 0 k b, < (x-k), oI , J (xy)dy >, (using (2.4))I (y) / J (ky)dy < This defines a regular distribution F(z) (kz n Since f is integrable over (O,b), its finite Hankel transform may be extended to the analytic function F(z) olbf(x) J-f J (zx)dx.
We show that F / (f).Since f is a regular generalized function, Since the integrand f(x)+(y) (xy);J (xy) is absolutely integrable over the domain 0 x b, 0 y , the order o integration may be changed, and we obtain /f'@ of dy (y) olbdx f(x) (xy)  where ai() is a constant depending on , for each i.If gi-r (x) xi-rh(x), then gi_r(X) is continuous on (0,) and gi_r(X) 0 outside (O,b).Since (y) is of rapid descent and (xy)1/2 Ju+i(xY) is bounded on 0 < xy (R), the order of integration in (4.3)may be interchanged.Therefore, (4.3)becomes From Example we know that / a(x-k) (kz)1/2J (kz) for 0 < k b.
Thus we get the same result as derived in Example 1.
Example 7.While calculating the finite Hankel transform of a(x-k), 0 < k b (Example 5) using the method of Theorem 4.2, it was necessary to evaluate certain integrals to find F(y).This may be avoided by using the above Corollary.From the definition of h(x), we see that D h(x)= I 0 : 1  Write n(t) --( + 1/2) -Clearly n(t) is a smooth function on (0,(R)) and n(t) O, for t b.Also, Yk (r,) y +i() < , for each k 0,1,2 Hence, n(t) Bu,b.Therefore, n(t) o(tU+1/2), as t 0+.
(5.8) ljll o x(O,b) Thus C O becomes a topological vector space We need the following lemma which is u,b stated without proof since the proof is identical to the proof of [5,Lemma I,p.
Now define hb(X) to be the periodic extension of period b of h(x) on R + (0 b] Then for any x hb(X) h(x-nb) for some positive integer n such that 0 x-nb b.Associated with hb(X) is the regular distribution in H (R)' (1) having the value The right-hand siae of (6.2) converges, since is of rapid descent as x .Now define a functional fb on Hu(1) by r --1/2 < fb' > (-l)r of(R) hb(X)( Dx) (x (x))dx So we see that (b s a periodic extension of f.Theorem 6.2.Every f in Bu,b may be extended to a periodic linear functional, H with period b, on u(1) which is continuous in the topology of Hu.
Theorem 6 3 For every in (0 b/4) and each f in B' Proof.The proof of the above theorem is similar to the proof of [1, Lemma 12] given by Zemanian.Theorem 6.4.The finite Hankel transform, huf, of a generalized function f in

.
For a given real number b 0 Y is the space of functions (z) which satisfy: z--(z) is an even entire function of z and for each non-negative integer k ),' d (xy)dy h, Ip[@].

4 ,
shall use only the weak topology of B' that is the topology ,b' assigned to it by the seminorms p (f) l<f,> B,b, f B' u,b" Since B is a sequentially complete countably normed space, Theorem 1.83].Similarly, we equip Y',b with the weak topology generated by the seminorms (F) I<F >I Y b' F Y' ,b" seen to define a generalized function f in B' On the other hand, if is an arbitrary testing function in D(

4 FINITE
HANKEL TRANSFORMATION OF B' Henceforth, we assume that e -.For f B' Bu and =,z Y ,b' ,b u ,b'we define the fnte Hankel transform F h f by F, < f, >.

Example 2 . 1 Example 4 .
The finite Ha[kel transform of 6(x-k) for k b is the "Zero" generalized function in Y' u,b since <6(x-k) (x)> (k) 0 for all in Bu,b.Example 3. The finite Hankel transform of the generalized function in B' defined ,b by (3.1) is the generalized function defined by the series F s yVT n J (y) Suppose f is a regulaF generalized function, corresponding to a Lebesgue integrable function over (O,b), in B' Than the ordinary finite Hankel transform ,b" of f is given by o 'b f(x) /Xn x J ( x)dx; n 1,2 3 converges in Y then the sequence of m ,b numbers { F( )nl(>, )} also converges Hence, the sum F(n)( defines a 2 n n n continuous linear functional on Y' ,b" Next we investigate a representation for the finite Hankel transform of a generalized function in B' Let D(O b) be the space of infinitely differentiable ,b" functions on (O,b) with compact support contained in (O,b).The topolog. of D(O,b) is that which makes its dual D'(O,b) of Schwartz's distribution.Then D(O,b)C B, b and h maps D(O b) into a subspace of Y Let W be the u,b" subspace of Y,b onto which D(O,b) is mapped.Then we have Theorem 4.2.For any generalized function f in B,b, there exists a continuous function F(y) of slow growth such that the finite Hankel transform ; f of f restricted to W is equivalent, in the functional sense, to the regular generalized function F in Y' ,b" Proof.For a given generalized function f, there exists an integer r 0 and a continuous function h(x) [5, Theorem 3.4.2]such that < f, > Drh, >, for every @ in D(O,b).We take h 0 outside (O,b).Then using (2.4), we have, for D(O,b), ) r h(x), Dr@(x) (-I) r I b dx h(x) Dr(x) o )r (-l)r o Ib dx h(x) o I dy (y)@-( x/y d(xy)) x) o I dy (y) 11 [(-l)iai(,)yixi-r x/'y a,+i(xy)] i+ra (,) dx xi-rh(x)[ol=dy ,y,yif x a (xy)] (4 3) i=o +i l)i+rai() I dy (y)y hu+i[gi_r(X)].i=o +i Denote /+i[gi_r(X)] by Gi_r(y), then for Wi+rai()yiGi_r i+u (y) <, it is obvious that F(y) is of slow growth.

FY' p b
restricted to a larger subspace than W of Y This section is very similar to section 3.4 of Zemanian[5, p. 86-93], consequently the corresponding results will be stated without proof or, perhaps, with only an indication of the proof.To begi with, we define certain spaces associated with B Definition 5.1.We define the spaces Bp b' Cp b and B((t Dt)[t-u-O(t) t-.-I [_(.+ ) 88].o o Lemma 5.3.Bu,b is a dense subset of Cu,b-The followip.gproposition gives an integral representation for the functions in

Proof. Trivial.
that generalized functions in B' of certain continuous functions.are distributional derivatives (5.13)We now extend f, satisfying the inequality (5.13), continuously and uniquely onto the space C O Let g in C O be arbitrary.Then by Lemma 5 3 there exists a u,b" u,b sequence {n of testing functions in B u,b such that Cn converges to g in Suppose f e B'

LeFmma 5 . 9 .
Let o be a fixed testing function in B u,b satisfying(5.23).where n is in H (I) u,b and the constant K is given by K o Ib t-u+1/2(t)dt.

( 5 .
25) Suppose f is a regular generalized function in B' generated by a differentiable function f such that f is Lebesgue integrable over (O,b) and f' is bounded on (O,b].Then for n H (I) u,b we have <f,,> o Ib f(x) n (x)dx o Ib xU-1/2f(x) [x-U-}@(x)]dx (for some B (I)) + b (x-]f(x)) (x)dx,