Distributional Hankel type transformation and Hankel type convolution.

1. Introduction

The Hankel transformation and its theory has been studied by many researchers, mathematicians such as Betancor and Marrero [1, 2, 3, 4], Dube and Pandey [6], Koh and Li [11], Koh and Zemanian [12], Zemanian [15, 16, 17] etc.

The Hankel type transformation is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [J.sub.[alpha]-[beta]] denotes the Bessel function of the first kind and order ([alpha] - [beta]) > 1/2.

We introduce the space [H.sub.[alpha],[beta]] of all complex-valued functions f on I = (0, [infinity]) such that

[[rho].sup.[alpha],[beta].sub.k,m](f) [=.sup.sup.sub.x[member of](0, [infinity])] [absolute value of [(1 + [x.sup.2]).sup.k] [(1/x D).sup.m] ([x.sup.2[beta]-1] [phi](x))] < [infinity], (1.2)

for every m, k, [member of] N. We note that the space [H.sub.[alpha],[beta]] is a Frechet Space.

The generalized Hankel type transform [h'.sub.[alpha][beta]] f of f [member of] [H'.sub.[alpha],[beta]] is defined by

<[h'.sub.[alpha],[beta]]f, [phi]> = <f, [h.sub.[alpha][beta]][phi]>, [phi] [member of] [H.sub.[alpha],[beta]]. (1.3)

For a > 0, we define the space [B.sub.[alpha][,[beta],a] as the space of those functions [phi] [member of] [H.sub.[alpha],[beta]] p such that [phi](x) = 0 for every x [greater than or equal to] a. It is equipped with the topology induced on it by [H.sub.[alpha],[beta]]. Note that it is a Frechet space too. The space [B.sub.[alpha],[beta]] is the inductive limit of the family [{[B.sub.[alpha],[beta],a]}.sub.a<0]. It is clear that if 0 < a < b then [B.sub.[alpha],[beta],a] [subset] [B.sub.[alpha],[beta],b]. Following Hirschman [10] and Haimo [9], we can study convolution for a Hankel type transformation, which is closely connected with, [h.sub.[alpha],[beta]] and one can deduce analogous results for the Hankel type transformation [h.sub.[alpha],[beta]]. We study the Hankel type convolution over the space [L.sub.[alpha],[beta],1] of measurable functions [phi](x), x[member of] (0, [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.1: For x, y, z[member of] (0, [infinity]), we define [D.sub.[alpha],[beta]] (x, y, z) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Definition 1.2: Let [phi][member of] [L.sub.[alpha],[beta],1] and x[member of] (0, [infinity]). Then the Hankel type translation operator is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

Definition 1.3: Let [phi], [phi][member of] [L.sub.[alpha],[beta],1] The Hankel type convolution is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

The Hankel convolution has been investigated on the spaces [B'.sub.[mu]] and [H'.sub.[mu]] of generalized functions in a series of papers by Betancor and Marrero [1-4]. We denote here the space of multipliers of [H.sub.[alpha],[beta]] and [H'.sub.[alpha],[beta]] by M and introduce the space [M'.sub.[alpha],[beta],#] = [h'.sub.[alpha],[beta]] ([x.sup.2[alpha]]M) [subset] [H'.sub.[alpha],[beta]] of convolution operators in [H.sub.[alpha],[beta]] and [H'.sub.[alpha],[beta]]. If f [member of] [H'.sub.[alpha],[beta]] and g[member of] [M'.sub.[alpha],[beta],#], then the Hankel type convolution f # g is the element of [H'.sub.[alpha],[beta]] defined by

<f # g, [phi]> = <f (x), <g(y), ([[tau].sub.x] [phi]) (y)>>, [phi][member of] [H.sub.[alpha],[beta]].

We state the following useful property of [D.sub.[alpha],[beta]](x, y, z):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

The space [M'.sub.[alpha],[beta],#] is a subspace of [H'.sub.[alpha],[beta]] that is closed under #--convolution. We shall need the following interchange formula of #--convolution in the sequel, which is stated as:

If f [member of] [H'.sub.[alpha],[beta]] and g[member of] [M'.sub.[alpha],[beta],#], then

[h'.sub.[alpha],[beta]] (f # g) = [x.sup.2[beta]-1] [h'.sub.[alpha],[beta]] (f)[h'.sub.[alpha],[beta]](g). (1.8)

Inspired by the work of Betanor & Gonzalez [5], we investigate the Hankel type convolution in a new subspace of [H'.sub.[alpha],[beta]]. We consider a Frechet space [H.sub.[alpha],[beta],k] of functions such that [M'.sub.[alpha],[beta],#] [subset] [H'.sub.[alpha],beta],k] k [member of] Z, k < 0.

We shall require the following in the sequel, which can be proved easily.

Theorem 1.4: Let f, g[member of] [H'.sub.[alpha],[beta]k] and k[member of] Z,k < 0.. The Hankel type convolution f # g defined by <f # g, [phi]> = <f(x), <g(g), ([[tau].sub.x] [phi])(y)>>, [phi][member of] [H.sub.[alpha],[beta]] is an element of [H'.sub.[alpha],[beta],k].

Moreover, if f, g, [member of] [H'.sub.[alpha],[beta],k] then

(i) [h'.sub.[alpha],[beta]](f # g)(y) = [h'.sub.[alpha],[beta]](f)(y)[h'.sub.[alpha],[beta]](g) [y.sup.2[beta]-1], y [member of] I.

(ii) f # g = g # f.

(iii) f #(g # h) = (f # g)# h.

(iv) The functional [F.sub.[alpha],[beta]] defined by <[F.sub.[alpha],[beta]], [phi]) = [2.sup.[alpha]-[beta]] [GAMMA][(3[alpha] + [beta]).sup.lim.sub.x[right arrow]0+] [x.sup.2[beta]-1] [phi](x), [phi][member of] [H.sub.[alpha],[beta],k], is in [H'.sub.[alpha],[beta],k] and [F.sub.[alpha],[beta]] # f = f # [F.sub.[alpha],[beta]] = f.

(v) [[DELTA].sub.[alpha],[beta]] (f # g) = ([[DELTA].sub.[alpha],[beta]] f) # g = f # ([[DELTA].sub.[alpha],[beta]] g), where [[DELTA].sub.[alpha],[beta]] is the Bessel type operator [x.sup.2[beta]-1] D[x.sup.4[alpha]] D[x.sup.2[beta]-1] with D = d/dx.

2. The generalized Hankel type transformation

In this section we study the Hankel type transformation on a certain space of generalized functions by using the Kernel Method. The techniques and arguments employed here are usual in other studies on distributional integral transforms (see [6], [12], [11], [15]). Therefore the proofs of some of our results will be just outlined.

Let k[member of] Z, k < 0. We define the space [G.sub.[alpha],[beta],k] of complex valued smooth functions, [phi](x); x[member of] (0, [infinity]), such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every n [member of] N.

The space [G.sup.[alpha],[beta],k] is endowed with the topology generated by the family [{[[eta].sup.n.sub.[alpha],[beta],k]}.sub.m[member of][??]] D of seminorms. One can easily prove that [G.sub.[alpha],[beta],k] is a complete and hence a Frechet space. From Koh and Zemanian [12, (9)], we can infer that [B.sub.[alpha],[beta]] [subset] [G.sub.[alpha],[beta],k]. Denote by [H.sub.[alpha],[beta],k] the closure of [B.sub.[alpha],[beta]] in [G.sub.[alpha],[beta],k]. Thus [H.sub.[alpha],[beta],k] is also a Frechet space. Note that the space [H.sub.[alpha],[beta],k] does not coincide with [G.sub.[alpha],[beta],k]. Let [[phi].sub.k](x) = [x.sup.2[alpha]][(1 + [x.sup.2]).sup.-k], x[member of] I. Following Koh and Zemanian [12, (9)], we have for every n [member of] N,

[[DELTA].sup.n.sub.[alpha],[beta]] [phi](x) = [x.sup.2[alpha]] [n.summation over (i=0)] [b.sub.i,n] [x.sup.2i][(1/x D).sup.n+i] [[x.sup.2[beta]-1] [phi](x)],

where [b.sub.i,m], i = 0,....., n, are suitable real numbers.

Thus

[x.sup.2[beta]-1] [[DELTA].sup.n.sub.[alpha],[beta]] [[phi].sub.k](x) = [n.summation over (i=0] [b.sub.i,n] [2.sup.n+i] (-k) (-k - 1)...(-k - n - i + 1)[(1 + [x.sup.2]).sup.-k-n-i], x[member of] I.

Thus [[eta].sup.n.sub.[alpha],[beta],k]{[[phi].sub.k]) < [infinity], n [member of] N and [[phi].sub.k][member of] [G.sub.[alpha],[beta],k]. On the other side, if [[phi].sub.k] is in [H.sub.[alpha],[beta],k], then there exists a sequence [([[phi].sub.k,m]).sub.m[member of]N]. [subset] [B.sub.[alpha],[beta]] with [[phi].sub.k,m] [right arrow] [[phi].sub.k] in [G.sub.[alpha],[beta],k] as. m [right arrow] [infinity].

More specifically.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence there exists [m.sub.0] [member of] N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x [greater than or equal to] C, with some constant C > 0, because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction.

Hence [[phi].sub.k] [not member of] [H.sub.[alpha],[beta],k].

Lemma 2.1 (sufficient condition): Let [phi][member of] [G.sub.[alpha],[beta],k]. If for each n[member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [phi][member of] [H.sub.[alpha],[beta],k].

Proof: Let r be a smooth function on I such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Define, for every m[member of] N-{0}, [r.sub.m](x) = [lambda](x - m + 1), x[member of] I, [[phi].sub.m](x) = [r.sub.m](x)[phi](x), x[member of] I.

By assumption [[phi].sub.m] [member of] [B.sub.[alpha],[beta]], m[member of] N. Now from Koh and Zemanian [12, (9)], we have for every n[member of] N, m[member of] N-{0} and x[member of] I,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [b.sub.i,n], i = 0,1,....., n are suitable real numbers.

Further for each p[member of] N and m[member of]N-{O}

[(1/x D).sup.p] ([[lambda].sub.m] (x) - 1) = [p.summation over (s=0)] [c.sub.s][x.sup.-2p+s] [D.sup.s]([[lambda].sub.m](x) - 1), x[member of] I,

where [c.sub.s], s = 0,1,....;p are certain real numbers.

Therefore there exists a constant K > 0 such that for each n[member of] N-{0} and x[member of] I,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [member of] > 0. Then there exists L > 0 such that

[absolute value of [(1 + [x.sup.2]).sup.k]] [x.sup.2[beta]-1][[DELTA].sup.n.[alpha],[beta]] [[[phi].sub.m](x) - [phi](x)]] < [epsilon], x [greater than or equal to] L, m[member of] N-{0}.

Also as [r.sub.n]{x) = 1 for x[member of] (0,m) and m[member of] N-{0}, there exists [m.sub.0] [member of] N-{0} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that [[phi].sub.m] [right arrow] [phi] in [G.sub.[alpha],[beta],k] as n [right arrow] [infinity]. Hence [phi][member of] [H.sub.[alpha],[beta],k]. This completes the proof.

Remark: From Lemma 2.1, one can immediately deduce that [H.sub.[alpha],[beta]] [subset] [H.sub.[alpha],[beta],k].

Theorem 2.2: (Application of Lemma 2.1): Let y[member of] I and k[member of] Z, k < 0. The function

[[phi].sub.y](x) = [(xy).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]](xy), x[member of] I, is in [H.sub.[alpha],[beta],k].

Proof: Let n[member of] N. An application of Zemanian [17, Lemma 5.4-1(5)] give us

[[DELTA].sup.n.sub.[alpha],[beta],k] ([(xy).sup.[alpha]+[beta]] [J.sup.[alpha]-[beta]](xy)) = (-[y.sup.2]) [(xy).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]](xy), x[member of] I.

As [z.sup.-([alpha]-[beta])] [J.sub.[alpha]-[beta]](z) is a bounded function on I, there exists K > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [[phi].sub.y] [member of] [G.sub.[alpha],[beta],k].

But by using Zemanian [17, ch.5, (6)], for every n[member of] N, we have

[x.sup.n] [(1/x D).sup.n] ([x.sup.-2[beta]-1] [[phi].sub.y](x)) = [(-1).sup.n] [y.sup.2[alpha]+n][(xy).sup.-([alpha]-[beta])] [J.sub.[alpha]-[beta]+n] (xy), x[member of] I.

Therefore, for each n[member of] N.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus from Lemma 2.1, we can infer that [[phi].sub.y] [member of] [H.sub.[alpha],[beta]]. This completes the proof.

In the following theorem, we prove that [[DELTA].sub.[alpha],[beta]] defines a continuous linear mapping from [H.sub.[alpha],[beta],k] into itself.

Theorem 2.3: Let k[member of] Z, k < 0 and let P be a polynomial. Then the mapping [phi] [right arrow] p([[DELTA].sub.[alpha],[beta]]) [phi] is linear and continuous from [H.sub.[alpha],[beta],k] into itself.

Proof: Enough to show that [[DELTA].sub.[alpha],[beta]] defines a continuous linear mapping from [H.sub.[alpha],[beta],k] into itself. Let [phi][member of] [H.sub.[alpha],[beta],k] Then there exists a sequence [([[phi].sub.m]).sub.m[member] of]N] [B.sub.[alpha],[beta]] such that [[phi].sub.m] [right arrow] [phi] in [H.sub.[alpha],[beta],k] as m [right arrow] [infinity]. One can immediately infer that [([[DELTA].sub.[alpha],[beta]] [[phi].sub.m]).sub.m[member of]N]. [subset] [B.sub.[alpha],[beta]] But for each n[member of]N and [phi][member of] [G.sub.[alpha],[beta],k],

[[eta].sup.n.sub.[alpha],[beta],k] ([[DELTA].sub.[alpha],[beta]] [phi]) = [[eta].sup.n+1.sub.[alpha],[beta],k] ([phi]).

Therefore it is clear that

[[DELTA].sub.[alpha],[beta]] [[phi].sub.m] [right arrow] [[DELTA].sub.[alpha],[beta]] [phi] in [H.sub.[alpha],[beta],k], as m [right arrow] [infinity]

and the mapping [phi] [??] [[DELTA].sub.[alpha],[beta]] [phi] is continuous. This completes the proof.

From Betancor and Marrero [3], it is clear that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also from Koh and Zemanian [12, (9)], it follows that [H'.sub.[alpha],[beta],k] [subset] [H'.sub.[alpha],[beta]].

We require the following space in the sequel.

Now we denote [M.sub.[alpha],[beta],k] as the space of all those locally integrable functions on (0, [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [M.sub.[alpha],[beta],k] [subset] [H'.sub.[alpha],[beta],k].

Theorem 2.4: Let k [member of] Z,/c< 0, and let P be a polynomial. Then the mapping f [??] P([[DELTA].sub.[alpha],[beta]])f is linear and continuous from [H'.sub.[alpha],[beta],k] into itself when in [H'.sub.[alpha],[beta],k] we consider either the weak or the strong topology.

Proof: Proof follows from Theorem 2.3.

Before establishing some properties of the generalized Hankel type transformation, we require the definition:

For every f [member of] [H'.sub.[alpha],[beta],k], the generalized Hankel type transformation [H'.sub.[alpha],[beta],k] is defined

as

([h'.sub.[alpha],[beta]]f)(y)= <f (x),[(xy).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]](xy) >, x[member of] I.

Theorem 2.5: Let k[member of]X, k<0, and let P be a polynomial. Then for every f[member of] [H'.sub.[alpha],[beta],k], we have

[h'.sub.[alpha],[beta]] [P([[DELTA].sub.[alpha],[beta]]) f](y)=p (-[y.sup.2]) [h'.sub.[alpha],[beta]](f)(y), y [member of] I.

Proof: Proof follows by using the fact that

[[DELTA].sub.[alpha],[beta]] [z.sup.[alpha]-[beta]](z) = [z.sup.[alpha]-[beta]] (z)

(refer to Zemanian [17, Ch. 5, (6), (7)]).

Theorem 2.6: Let k [member of] Z, k < 0, and f [member of] [H'.sub.[alpha],[beta],k]. There exists K > 0 and t[member of]N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof: Using Theorem 1.8-1 and relations (6) and (7) of chapter 5 in Zemanian [17], proof can be completed.

Theorem 2.7: Let k[member of]Z,k<0, and f[member of] [H'.sub.[alpha],[beta],k]. Then [h'.sub.[alpha],[beta]] f is (-2k-1) times differentiable.

Proof: We prove first that [h'.sub.[alpha],[beta]] f is continuous in I. For every ye I and 0 < [absolute value of h]< y, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we can show that

[(x (y + h)).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]] (x (y + h))[right arrow][(xy).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]](xy) (2.1)

in [H.sub.[alpha],[beta],k] as h [right arrow] 0, then the continuity of f in y[member of] I will be established. So we first prove (2.1).

For, let y[member of] I, and n[member of] N, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now let [member of] > 0. As [z.sup.-([alpha]-[beta])] [J.sub.[alpha]-[beta]](z) is bounded on I there exists L>0 such that for x [greater than or equal to] L and 0 <[absolute value of h]< y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Now by using mean value theorem we can find [h.sub.0] > 0 such that for every 0 < x < L and 0<[absolute value of h] <[h.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Using (2.2) and (2.3) we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided that 0<[absolute value of h] <[h.sub.0]. This proves (2.1).

Next we prove that [h'.sub.[alpha],[beta]] f is differentiable provided k [member of] Z, k [less than or equal to] -1. Let 0 < y < [infinity]. For 0 < [absolute value of h] < y, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

.It will be established that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in [G.sub.[alpha],[beta],k] as h [right arrow] [0.sup.+].

Now for every 0 < [absolute value of h] < y and 0 < x < [infinity], we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let n[member of]N. For every x[member of]I and 0 < [absolute value of h] < y, from Zemanian [17, Ch. 5, (6), (7)] we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As [z.sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]] (z) is bounded on I, there exists a constant K > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, for x[member of] I and 0 < [absolute value of h] < y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

provided that K [less than or equal to] 1.

Hence, [h'.sub.[alpha],[beta],k] is differentiable when k [member of] Z, k [less than or equal to] - 1. By using the similar argument, the proof of general case follows. Thus proof is complete.

Theorem 2.8: Let k[member of]Z,k<0, and f[member of] [H'.sub.[alpha],[beta],k]. Then <[H'.sub.[alpha],[beta]]f,[phi]> = <f, [h.sub.[alpha],[beta]][phi],>, [phi][member of] [H'.sub.[alpha],[beta]].

Proof: Proof can be given by proceeding as in the Theorem 3 of Koh and Zemanian [12] and replacing the function [e.sup.-[alpha]x] ([alpha] > 0) by [(1 + [x.sup.2]).sup.k].

Theorem 2.9: Let k[member of]Z,k< 0, and f,g[member of] [H'.sub.[alpha],[beta],k]. If [h'.sub.[alpha],[beta]] f = [h'.sub.[alpha],[beta]] g then f = g.

Proof: Let [phi][member of] [H'.sub.[alpha],[beta],k]. There exists a sequence [([[phi].sub.m]).sub.me[??]] [subset] [B.sub.[alpha],[beta]] such that [[phi].sub.m] [right arrow] 0 in

[G.sub.[alpha],[beta],k] as m [right arrow] [infinity]. Then as f, g [member of] [H'.sub.[alpha],[beta],k,] we have

<f,[[phi].sub.m]> [right arrow] <f,[phi]> and <g,[[phi].sub.m]> [right arrow]<g,[phi], as m [right arrow] [infinity].

Now by Zemanian [17, Theorem 5.4-1] and Theorem 2.8, we can obtain

<f,[[phi].sub.m]> = <[h'.sub.[alpha],[beta]]f,[h.sub.[alpha],[beta]][[phi].sub.m]> = <[h'.sub.[alpha],[beta]]g,[h.sub.[alpha],[beta]][[phi].sub.m] = <g,[[phi].sub.m]>, m[member of]N.

Thus proof is complete.

Remarks: (i) Theorem 2.8 gives a uniqueness result for the generalized Hankel type transformation on [H'.sub.[alpha],[beta],k].

(ii) From Theorem 2.8 and 2.9 it follows that each generalized function f [member of] [H'.sub.[alpha],[beta],k] is uniquely determined by its Hankel type transform [h'.sub.[alpha],[beta]] f.

The Hankel type convolution on [H'.sub.[alpha],[beta],k]

In this section we define the Hankel type convolution on the spaces [H'.sub.[alpha],[beta],k] and analyze the Hankel type translation [[tau].sub.x], x[member of] I on [H'.sub.[alpha],[beta],k]. First, we establish that the operators [[DELTA].sub.[alpha],[beta]] and [[tau].sub.x], x[member of] I commute.

Lemma 3.1: Let n[member of] N and k[member of]Z,k<0, Then for every [phi][member of] [H.sub.[alpha],[beta],k],

[[DELTA].sup.n.sub.[alpha],[beta],x]([[tau].sub.x][phi])(y) = [[tau].sub.x]([[DELTA].sup.n.sub.[alpha],[beta]][phi])(y),x,y[member of]I.

Proof: For [phi][member of] [H.sub.[alpha],[beta],k], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Let q>0. Consider a smooth function r on (0, [infinity]) such that r(x) = 1 for x[member of] (0,2q) and r (x) = 0 for x[member of](2q +1, [infinity]). Now we prove that r[phi][member of][B.sub.[alpha],[beta]]. In fact we consider the vector space.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Following usual techniques, it can be proved that [N.sub.[alpha],[beta]] endowed with the topology generated by the family [{[[eta].sub.n]}.sub.n[member of]N] of seminorms is a Frechet space. If [([[phi].sub.m]).sub.m[member of]N] [subset] [B.sub.[alpha],[beta]] [subset] [N.sub.[alpha],[beta]] is such that [[phi].sub.m] [right arrow] [phi][member of][G.sub.[alpha],[beta],k] as m [right arrow] [infinity], according to Sanchez [13, Ch. IV, proposition 2] and by using the Leibniz formula we can find K such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

p, q[member of]N. Thus [(r[[phi].sub.m])m[member of]N] is a Cauchy sequence in [N.sub.[alpha],[beta]]. Then there exists [psi][member of] [N.sub.[alpha],[beta]] such that r[[psi].sub.m] [right arrow] [psi] in [N.sub.[alpha],[beta]] as n [right arrow] [infinity]. Thus since r(x)[[phi].sub.m](x)[right arrow](x) and [(1 + [x.sup.2]).sup.k] [x.sup.2[beta]-1] [[phi].sub.m](x) [right arrow] [(1 + [x.sup.2]).sup.k] [x.sup.2[beta]-1] [phi](x) for x[member of] I as m [right arrow] [infinity], it follows that r[phi] = [psi] and therefore we have r[psi][member of] [B.sub.[alpha],[beta]].

On the other side, since [phi](y) = ([phi]r)(y) for 0<y<2q, from (3.1) we can infer that

([[tau].sub.x] [phi])(y) = ([[tau].sub.x] [phi]r)(y) for x,y[member of](0,q).

Therefore by Betancor and Marrero [3, (1.3)], we conclude that for each n[member of]N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

0 < x, y < q. Thus as q>0 is arbitrary, the result is established. This completes the proof.

Theorem 3.2: Let k[member of]Z,k<0, For every x[member of]I, the mapping [phi] [??] [[tau].sub.x][phi] is linear and continuous from [H.sub.[alpha],[beta],k] into itself.

Proof: Let x[member of] I. From Corollary 3.3 of Betancor and Marrero [3], [[tau].sub.x][phi][member of] [B.sub.[alpha],[beta]] for every [phi][member of] [B.sub.[alpha],[beta]]. Let [phi][member of]. [H.sub.[alpha],[beta],k]. There exists a sequence [([[phi].sub.m]).sub.m[member of]N] [subset] [B.sub.[alpha],[beta]] such that [[phi].sub.m] [right arrow] [phi] [member of] [G.sub.[alpha],[beta],k] as m [right arrow] [infinity]. By Lemma 3.1 and Lemma 2.1 of Gonzalez and Negrin [7], for each n,m[member of]N we can, obtain for x, y [member of] I,

Now by using (1.7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Therefore [[tau].sub.x][[phi].sub.m] [right arrow] [[tau].sub.x] [phi] [member of] [G.sub.[alpha],[beta],k] as m [right arrow] [infinity] and [[tau].sub.x] [phi] [member of] [H.sub.[alpha],[beta],k]. Also an inequality similar to (3.2) proves that the Hankel type translation [[tau].sub.x] defines a continuous mapping from [H.sub.[alpha],[beta],k] into itself. This completes the proof.

Now we define the Hankel type convolution of a distribution in [H'.sub.[alpha],[beta],k] and a function in [H.sub.[alpha],[beta],k]. If f [member of] [H'.sub.[alpha],[beta],k] and [phi] [member of] [H.sub.[alpha],[beta],k] then the Hankel type convolution f # g is defined by

(f #[phi])(x) = <f,[[tau].sub.x] [phi], x[member of] I. (3.3)

Notice that if f[member of][G.sub.[alpha],[beta],k] then for each [phi][member of] [H.sub.[alpha],[beta],k],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that the classical #-convolution can be seen as a special case of the distributional #-convolution (3.3).

Theorem 3.3: Let k [member of] N, k < 0 and f [member of] [H'.sub.[alpha],[beta],k]. Then the mapping [phi][right arrow]f #[phi] is linear and continuous from [H.sub.[alpha],[beta],k] into itself.

Proof: We divide the proof in three cases:

Case I: For each [phi][member of][B.sub.[alpha],[beta]] and n[member of]N,

[[DELTA].sup.n.sub.[alpha],[beta],x] <f(y), ([[tau].sub.x] [phi])(y)> = <f(y), [[tau].sub.x]([[DELTA].sup.n.sub.[alpha],[beta],x] [phi])(y)>, x[member of]U. (3.4)

Let [phi] [member of] [B.sub.[alpha],[beta]]. By (1.3) of Betancor and Marrero [3], we have

([[tau].sub.x] [phi])(y) = [h.sub.[alpha],[beta]] [[t.sup.2[beta]-1][(xt).sup.[alpha]+[beta]][J.sub.[alpha]-[beta]](xt)h [alpha]-[beta]([phi](t)](y),x,y[member of] I. We shall prove that

[[DELTA].sub.[alpha],[beta],x] (f(y),([[tau].sub.x] [phi])(y)) ={f(y), [[DELTA].sub.[alpha],[beta],x] ([[tau].sub.x] [phi])(y)), x[member of]i. (3.5)

Firstly we must see that for each x [member of] I

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Let x[member of] I and 0 < [absolute value of h] < x. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then for n[member of] N and y[member of]I, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But as [z.sup.-([alpha]-[beta]))] [J.sub.[alpha]-[beta]](z) is bounded function on I and

d/dz[[z.sup.-([alpha]-[beta]))] [J.sub.[alpha]-[beta]](z)] = [z.sup.-([alpha]-[beta]))] [J.sub.[alpha]-[beta]+1](z),z[member of] 1,

we can deduce that

[(1 + [y.sup.2]).sup.k] [y.sup.2[beta]-1] [[DELTA].sup.n.sub.[alpha],[beta],y] [I.sub.h](y) [right arrow] 0 (3.7)

as h [right arrow] 0 uniformly in y[member of](0, [infinity]). Since f [member of] [H'.sub.[alpha],[beta],k], (3.6) follows from (3.7). By proceeding in the same manner, we can prove that

d/dx <f(y),[x.sup.4[alpha]] d/dx [x.sup.2[beta]-1] ([[tau].sub.x][phi])(y)> = <f(y), [d/dx] [x.sup.4[alpha]] [d/dx] [x.sup.2[beta]-1]([[tau].sub.x][phi])(y)>,x[member of] I.

Thus (3.5) is proved.

Now from Lemma 3.2 and (3.5), (3.4) can be proved immediately.

Case II: The mapping [phi] [??] f#[phi] is continuous from [B.sub.[alpha],[beta]] into [G.sub.[alpha],[beta],k] when we consider on [B.sub.[alpha],[beta]] the topology induced by [G.sub.[alpha],[beta],k].

Since f [member of] [H'.sub.[alpha],[beta],k], according to Zemanian [17, Theorem 1.8-1] there exists K > 0 and re such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

Let [phi][member of] [B.sub.[alpha],beta] and n[member of] N From (3.4) and (3.8) we can deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus by proceeding in a similar manner as in the proof of (3.2), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof of Case II.

Case III: For every [phi] [member of] [B.sub.[alpha],beta], we have f #[phi] [member of] [H.sub.[alpha],[beta],k]. Let [phi] [member of] [B.sub.[alpha],beta]. By Case II, f#[phi][member of][G.sub.[alpha],[beta],k]. To see that f#[phi][member of][H.sub.[alpha],[beta],k], we shall use Lemma 2.1 Let n[member of]N By using (1.2) of Betcancor and Marrero [3], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now by using Lemma 5.4-1 of Zemanian [17] for some K > 0 and r[member of] N we have, for x[member of] I,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[less than or equal to] K, because [z.sup.-([alpha]-[beta])] [J.sub.-[alpha]-[beta]](z) is bounded on I.

We can complete the proof by using the above cases. As the space [B.sub.[alpha],beta] is dense in [H.sub.[alpha],[beta],k] the mapping [B.sub.[alpha],beta] [right arrow] [H.sub.[alpha],[beta],k], [phi] [??] f #[phi], can be continuously extended to [H.sub.[alpha],[beta],k]. Let T be the extended mapping. We know that if [phi][member of] [H.sub.[alpha],[beta],k] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the limit is understood in [H.sub.[alpha],[beta],k] and [([[phi].sub.m]).sub.m[member of]N] is a sequence in [B.sub.[alpha],beta] such that [[phi].sub.m] [right arrow] [phi][member of][H.sub.[alpha],[beta],k] as m[right arrow][infinity]. It is not very difficult to see that convergence in [H.sub.[alpha],[beta],k] implies pointwise convergence on (0,[infinity]). However, by (3.2) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with K > 0 and r[member of]N.

Thus, (f #[[phi].sub.m])(x) [right arrow] (f #[phi])(x) as m [right arrow] [infinity] for every x[member of](0,[infinity]). Therefore, we conclude that (T[phi])(x) = (f #[phi])(x), x[member of] I and f #[phi] [member of] [H.sub.[alpha],[beta],k]. Thus the proof is complete.

Now we can define #-convolution in [H'.sub.[alpha],[beta],k] : If f,g,[member of] [H'.sub.[alpha],[beta],k] then Hankel type convolution f# g is defined as

<F#g,[phi]>=<f(x),<g(y),([[tau].sub.x][phi])(y)>>,[phi][member of][H.sub.[alpha],[beta],k].

By Theorem 3.3, f #g[member of] [H'.sub.[alpha],[beta],k]. Hence, the Hankel type convolution is a closed operation in [H'.sub.[alpha],[beta],k].

4. Algebraic properties of # -convolution:

In this section we establish main algebraic properties of #-convolution.

Theorem 4.1: Let k[member of]Z,k<0, If f,g,h[member of][H'.sub.[alpha],[beta],k]then:

(i) [h'.sub.[alpha],[beta]](f#g)(y) = [h'.sub.[alpha],[beta]](f)(y) [h'.sub.[alpha],[beta]](g)(y)[y.sup.2[beta]-1], y[member of]I.

(ii) f#g = g# f

(iii) f # (g#h) = (f#g)# h

(iv) The functional [F.sub.[alpha],[beta]] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[F.sub.[alpha],[beta]] # f = f # [F.sub.[alpha],[beta]] = f.

(v) [[DELTA].sub.[alpha],[beta]] (f # g) = ([[DELTA].sub.[alpha],[beta]] f) # g = f # ([[DELTA].sub.[alpha],[beta]]g)

Proof : (i) For each ye I, according to Watson [14, p.367 and p.411], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[[phi].sub.y] (x) = [(xy).sup.[alpha]+[beta]] [J.sub.[alpha]-[beta]](xy), x, y[member of] I.

(ii) Using (i) we can infer that

[h'.sub.[alpha],[beta]](f#g)(y) = [h'.sub.[alpha],[beta]](f)(y) [h'.sub.[alpha],[beta]](G)(y)[y.sup.2-[beta]-1] = [h'.sub.[alpha],[beta]](g#f)(y), y[member of]I.

Thus by Theorem 2.9, f # g = g # f.

(iii) This is just a consequence of property (i) and Theorem 2.9.

(iv) Let [phi] [member of] [H.sub.[alpha],[beta],k]. There exists a sequence [([[phi].sub.n]).sub.ne[??]] in [B.sub.[alpha],beta] such that [[phi].sub.n] [right arrow] [phi] in

[H.sub.[alpha],[beta],k] as n [right arrow] [infinity].

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus there exists q[member of]c such that [q.sub.n] [right arrow] q as n [right arrow] [infinity]. Moreover it is not hard to see that the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists and is equal to q.

Further we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [F.sub.[alpha],[beta]] [member of] [H.sub.[alpha],[beta],k].

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore by invoking (i) we obtain [h.sub.'[alpha],[beta]](f # F[alpha],[beta])= [h.sub.'[alpha],[beta]] (f). Then by Theorem

2.9, we have f # [F.sub.[alpha],[beta]] = f.

(v) Proof of this property follows from Theorem 2.9 and 2.5 by using property (i) again.

This completes the proof.

Theorem 4.2: (Continuity of #-convolution): Let k[member of]Z,k<0, Assume that [([f.sub.n]).sub.n[member of]N], is a sequence in [h.sub.'[alpha],[beta],k] that converges to f [member of] [h.sub.'[alpha],[beta],k] in the weak topology (respectively in the strong topology) of [h.sub.'[alpha],[beta],k]. Then for every g[member of][h.sub.'[alpha],[beta],k],

[f.sub.n] # g [right arrow] f # g as n [right arrow] [infinity]

in the weak topology (respectively in the strong topology of [h.sub.'[alpha],[beta],k]). Proof: Proof follows from Theorem 3.3.

References

[1] Betancor J.J. and Marrero I., 1992, "Multipliers of Hankel transformable generalized functions", Comment. Math. Univ. Carolin. 33, 389-401.

[2] Betancor J.J. and Marrero I., 1993, "The Hankel Convolution and the Zemanian spaces BM and B'M," Math. Nachr. 160, 277-298.

[3] Betancor J.J. and Marrero I.,1993, "Structure and convergence in certain spaces of distributions and the generalized Hankel convolution", Math. Japon. 160, 1141-1155.

[4] Betancor J.J. and Marrero I., 1993, "some properties of Hankel convolution operators", Canad. Math. Bull. 36, 398-406.

[5] Betancor J.J. and Marrero I.,1995, "A Hankel convolution operation for a distributional Hankel transformation", Studia Mathematica 117(1), 57-72

[6] Dube L.S. and Pandey J.N., 1975, "On the Hankel transform of distributions", Tohoku Math. J. 27, 337-354.

[7] Gonzalez B.J. and E.R. Negrin E.R, 1995, "Convolution over the spaces [S'.sub.k] ", J. Math. Anal. Appl. 190, 829-843.

[8] B.J.Gonzalez and E.R. Negrin, 1995, "Fourier transform over the spaces [S'.sub.k] ", ibid 194, 780-798.

[9] Haimo D.T., 1965, " Integral equations associated with Hankel convolutions", Trans. Amer. Math. Soc. 116, 330-375.

[10] Hirschman I.I. Jr.,1960/61, "Variation diminishing Hankel transforms", J.Anal.Math. 8, 307-336.

[11] Koh E.L.and Li C.K., 1992, "The complex Hankel transformation on M'M, Congr. Numer." 87, 145-151.

[12] Koh E.L. and Zemanian A.H., 1968, "The complex Hankel and I-transformations of generalized functions", SIAM J. Appl. Math. 16, 945-957.

[13] Sanchez A.M.,1987, "La transformacion integral generalizada de Hankel-Schwartz, Ph.D. Thesis", Dep. Analysis Matematico, Universidad de La Laguna,

[14] Watson G.N.,1958, "A Treatise on the theory of Bessel Functions", Cambridge University Press, London.

[15] Zemanian A.H., 1966, "The distributional Hankel transformation", SIAM J.Appl. Math. 14, 561-576.

[16] Zemanian A.H., "The Hankel transformation of certain distributions of rapid growth", ibid, 678-690.

[17] Zemanian A.H., 1968, "Generalized Integral Transformations", Interscience, New York, 1968.

B.B. Waphare *

* MAEER's MIT Arts Commerce and Science College, Alandi (D) Tal: Khed, Dist: Pune, pin 412105, (Maharashtra), INDIA
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