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Internat. J. Math. & Math. Sci.VOL. 14 NO. 2 (1991) 269-274

269

CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS

J.J. BETANCOR

Departamento de Anllsis MatemticoFaciltad de MatemticasUnlversidad de La Laguna

La Laguna, TenerifeCanary Islands, Spain

(Received August i, 1988 and in revised form April 30, 1989)

ABSTRACT. In this paper we prove a characterization theorem for the elements of the

space H’ of generalized functions defined by A.H. Zemanlan.

KEY WORDS AND PHRASES. Generalized functions, Hankel transforms.

1980 AMS SUBJECT CLASSIFICATION CODE. 46F12.

1. INTRODUCTION.

The Hankel transformation defined by

h {f(x)}(y)= f (xy)I/2j (xy)f(x)dx

where J denotes the Bessel function of the first kind and order , has been

extensively studied In recent years.

A classical result concerning the Hankel transformation is the following

inversion theorem (see [I]).

THEOREM I. Let f(x) (LI(0,) be of bounded variation in a neighborhood of the

point x x0. If >- and F(y) h{f(x)}(y), then

h-l{F(y)}(xO) f F(y)(XOY I/2j (x0Y)dy-1/2 {f(x0 + 0)+f(xO- 0)}.0

Another well known result is the Parseval’s equation (I) (see [I)).

THEOREM 2. Let f(x) and G(y) be elements of LI(0,). If F(y) and g(x) are

respectlvely the direct and inverse -th order Hankel transforms of f(x) and G(y),

then

/ f(x)g(x)dx / F(y)G(y)dy, for any0 0

270 J.J. BETANCOR

Other conditions under which Parseval’s equation holds are given by P. Macaulay-

Owen [2].

The h -transform has been extended to several spaces of generalized functions.

Apparently, A.H. Zemanian [I] was the first to extend the Hankel transform. He

introduced the space H of testing functions consisting of all infinitely

differentiable complex-valued functions defined on I=(0,) and such that

,nxl

for every m,nN. The Hankel transform is an automorphlsm onto H For every

f H’ (the dual space of H ), the generalized Hankel transformation H’f of f was

defined by the following generalization of Parseval’s equation

<h’f,> <f,h > for every H

h’ is an automorphism onto H’.

Later, E.L. Koh and A.H. Zemanian [3] defined the generalized complex Hankel

transformation. For a real number and a positive real number a the space J was

defined as the space of testing functions which are smooth on I and for which

xEl

where S x-- I/2 Dx2+IDx-- I/2. For each complex number y in the strip

R= C:llm yl<a, y e (- 0]} J contains the function (xy) J(xy). The

h -transform is now defined on the dual space ] as follows:

DEFINITION. Let be in the interval ( < -. Then, for every f e ’, and

y e R,

(h’f) (y)= <f(x), (xy)/2j (xy) >

E.L. Koh [4] showed that a distribution f ]’ can be written as a finite sum

of derivatives of continuous function of exponential descent. More specifically, he

established:

THOEREM 3 ([4]). Let f be ing’,. Then f is equal to a finite sum

kd i w-(ll2)-k+l

Ci(x) (e-ax x- Pi(x)Fi(x))i--0

where the Fi(x) are continuous on (0,-) and the P. (x) are polynomials of degree k.

Other Hankel ype transformations have been also extended to certain spaces of

generalized functions (see G. Altenburg [5], L.S. Dube and J.N. Pandey [6], J.M.

Mndez [7],...).

CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS 271

In this paper we pove a characterization theorem for the generalized functions

in H’ Our proof is anaogous to the method employed in structure theorems for

Schwartz distributions (see [8] and [4]).

In this paper a function (x) will be called of rapid descent if xmDq(x) tends

to zero, as x+, for every m,qeN.

2. The space H’ of generalized functions. A characterization theorem.

A useful result due to A.H. Zemanian (see [I]) is the following

PROPOSITION I. Let f be in H’.There exist a positive constant C and nonnegatlve

integers r,k such that

<f’>l C maX{Ym,nlJ (); Omr,Onk}, for every (H.We now present some new properties of the space H of testing functions.

m.1 n x--l/2PROPOSITION 2. Let be in H The function x (D) (x)) is

a) of rapid descent as x+=, and

b) in LI(0,-),for every m,neN.

PROOF. It is enough to take into account that

Ixm( D)n(x-u-I/2(x)) C xm,n

for every x I and m,

neN, C being a suitable positive constant.m,n

The main result of this paper is the next.

THEOREM 4. A functional f is in H’ if and only if, there exist bounded

measurable functions gm (x) defined on I, for m--0,1,...,r and n=0,1,...,k, where r

and k are nonegative integers depending onf, such that

r,k x-U- I/2<f,>=< Z (-lxl)n{xm(-D)gm,n(X)},(x)> (2.1)

m,n

for every H

PROOF. Let f be in H’. In veiw of Proposition I, there exist a constant C > 0

and nonnegative integers r and k depending on f such that

U (); 0 m ( r,0(nk}<f,> ( C maX{Ym, n

=Cmax {sup Ixm(1-D" n x-U- I/(x) I;x O(m(r,0(n(k},x61

for every H

272 J.J. BETANCOR

Since xmiD)’ln(x-U-I/2 @(x)) is of rapid descent as x+ (Proposition 2), we get

for every H m,n,eN.

Hence

where II ILl(O,. denotes the norm on the space LI(O,). Then we can write

t<f,,>lCmax{tlD%{tm(tl--D)n(t-U-l(t))}l Ll(O,.);Omr, Onk}

for every eH

We now define the injective map

F: H FH

(Dt{tm(It--D)n(t-- I/2 (x))})m=O,..., rn=O,... ,k

If FH is endowed with the topology induced in it by the product space

A (0 )=(LI(0 ))(r+l)(k+l) thenr,k

G:FH C

F <f, @>

is continuous linear mapping.

By application of the Hahn-Banach Theorem, G can be extended to A k(0 ).(r+l)(k+l) r,

Therefore, since A’ (0 ) is isomorhic to (L (0 )) (see F. Treves [I0])r,k

(m--0, ,r; n=0,.. ,k) suchthere exist (r+l) (k+l) bounded measurable functions, gm,nthat

r km #(x))}>=(x) D{x (D)n x--I/2G(F) <f > <gm,n

m=O, n=Or k -DI)n [xm(-D)g

m(x)} (x)>< x-U-I/m(

x ,nm=O, n=O

for every e H

CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS 273

On the other land, if is defined by (2) then ell’.

To see this, it is enough to prove that If_{@}veN, is a sequence in H such

xI_ /2,vkX)}vN to zerothat v 0 as v , then the sequence {xm( D)n( converges

as v , in Ll(O,=), for every re,heN. This completes the proof of the theorem.

The Hankel-Schwartz transfqrm defined by the pair

F(y)=B f(x)}(y) f x2V+Ib (xy)f(x)dxV 0 V

f(x)=B {F(y)}(x) f y2+Ib (xy)F(y)dyV 0

for - , where b (z) z-J (z) and J denotes the Bessel function of the first

kind and order , was introduced by A.L. Schwamtz [9], who estbalished its inversion

formula. This integral transfommatlon has been extended by G. Altenburg [5] and J.M.

Mendez [7] to the space H’II’of generalized functions (H=HI/2 [n their notation)

following a procedure analogous to the one employed by A.H. Zemantan [I]. By

setting V =-, we can deduce from Theorem 4 the next

COROLLARY. The functional f is in H’ if and only If, there exist bounded

measurable functions gm,n(X) defined on I, for m=O,...,r,n=O,...,k where r and k are

nonnegattve integers depending on f, such that

r k(x)},(x)>, H.<f,> < (-)n[xm(-D)gm,n

m=O, n=O

REFERENCES

I. ZEMANIAN, A.H. A distributional Hankel transformation, J. SIAM Appl. Math.

14(3) (1966), 561-576.

2. MACAULAY-OWEN, P. Parseval’s theorem for Hankel transform, Proc. London. Math.

Soc. 45 (1939), 458-474.

3. KOH, E.L. and ZEMANIAN, A.J. The complex Hankel and 1-transformatlons of

generalized functions, SIAM J. Appl. Math. 16 (1968), 945-957.

4. KOH, E.L. A representation of Hankel transformable generalized functions, SlAM

J. Math. Anal. (1970), 33-36.

5. ALTENBURG, G. Bessel transformationen in raumen von grundfunktlonen uber dem

interval =(0,)und derem Dualraumen, Math. Nachr. 108 (1982), 197-218.

6. DUBE, L.S. and PANDEY, J.N. On the transform of distributions, Tohoku Math.Journal 27 (1975), 337-354.

7. Mendez, J.M. On the Bessel transformation of the arbitrary order, Math. Nachr.136 (1988), 233-239.

8. TREVES, F. Topological Vector Spaces, Distributions and Kernels, Academic Press,New York, 1967.

274 J.J. BETANCOR

9. SCHWARTZ, A.L. An inversion theorem for Hanke[ transform, Proc. Amer. Math.Soc. 22 (1969), 713-717.

I0. SCHWARTZ, L. Theorie des Distributions, Vols. 1 and I[. Hermann, Paris, 1957and 1959.

GRAY, A., MATHEWS, G.B. and MACROBERT, T.M. A treatise on Bessel functions andtheir applications to Physics, McMillan and Co. Ltd., London, 1952.

12. WATSON, G.N. Theory of Bessel functions, Cambridge Universlty Press,Cambridge, 1958.

13. ZEMANIAN, A.H. Generalized Integral Transformations, Interscience Publishers,New York, 1968.

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