International Journal of Pure and Applied ————————————————————————– Volume 55 No. 2 2009, 247-256

THE HANKEL OF ARBITRARY ORDER

C.K. Li Department of Mathematics and Computer Science Brandon University Brandon, Manitoba, R7A 6A9, CANADA e-mail: [email protected]

Abstract: Let > 1/2. The classical Hankel transform is defined by − ∞ (hµφ)(t)= xJµ(xt)φ(x)dx t (0, ), (1) Z0 ∈ ∞ where Jµ(x) denotes the of the first kind and order . The goal of this paper is to construct the Hankel transform of arbitrary order hµ,k based −1 on the two differential operators and show that hµ,k = h on H 1 for R. µ,k µ− 2 ∈ Furthermore, the Hankel convolution of arbitrary order is introduced with the following identity −µ (hµ,kh)(t)= t (hµ,kφ)(t)(hµ,kψ)(t) on the spaces (H 1 ,H 1 ) and (Sµ,H 1 ) respectively. µ− 2 µ− 2 µ− 2

AMS Subject Classification: 46F12 Key Words: Hankel transform, Bessel function, Hankel convolution and automorphism

1. Introduction

Let R+ = (0, ) and a weight function ν(x) > 0 in R+. We define a set of ∞ functions L(R+,ν(x)) as ∞ L(R+,ν(x)) = f(x) f(x) ν(x)dx < . { | Z0 | | ∞} Different types of the Hankel transforms as well as their , including modified ones, have been investigated over the decades (for example, see [7]-

Received: August 1, 2009 c 2009 Academic Publications 248 C.K. Li

[5] and [1]) both in the classical sense and in spaces of generalized functions. These transforms are applied to solve problems of mathematical physics and differential equations with variable coefficients. In 1995, Tuan and Saigo [12] used the following commutative Hankel convolution h(x) of a function f(x) with a function g(x) (due to Zhitomirskii, see [14]), 21−3µx−µ h(x)= [x2 (u v)2]µ−1/2 √πΓ( + 1/2) ZZ u+v>x, |u−v| 1/2 and f(x), g(x) L(R+, √x). Then the ∈ function h(x) of (2) exists and the convolutional identity −µ (hµh)(t)= t (hµf)(t)(hµg)(t) holds. Recently, Britvina studied some polyconvolutions of the Hankel transform in [2], where she defined the following functions h1(t) and h2(t), and obtained Theorem 1.2 and Theorem 1.3, ∞ t+v µ−1 −µ+1 h1(t)= t dv u f(u)g(v)P1(t; u, v)du Z0 Z|t−v| ∞ ∞ µ−1 −µ+1 t dv u f(u)g(v)Q1(t; u, v)du, − Z0 Zt+v −µ µ h2(t)= t u f(u)g(v)P2(t; u, v)dudv ZZ|t−v|

√2 (2 1)π µ 1/2−µ µ−1/2 Qj(t; u, v)= sin[(ν )π]exp − v Q (chrj)sh rj, π3/2 −  2j  ν−1/2 µ µ j = 1, 2, and Pν , Qν are the associated Legendre functions of the first and the second kind, respectively, and 2tv cos s = t2 + v2 u2, 2tvchr = u2 t2 v2, 1 − 1 − − 2uv cos s = u2 + v2 t2, 2uvchr = t2 u2 v2. 2 − 2 − − Theorem 1.2. Suppose that f(t), g(t) L(R+, √t) and Re > 1/2, ∈ THE HANKEL CONVOLUTION OF ARBITRARY ORDER 249

Re ν > Re 1. Then the function h1(t) exists and the relation − −ν (hν h1)(x)= x (hµf)(x)(hν g)(x) holds. Theorem 1.3. Assume that f(t), g(t) L(R+, √t) and Re > 1/2, ∈ Re ν > (2 Re 3)/4. Then the function h (t) exists and the following re- − 2 lation −µ (hµh2)(x)= x (hν f)(x)(hν g)(x) is satisfied. Inspired by the studies of Gonz´alez and Negrin ([3] and [4]) on the convolu- tion and , Betancor and Gonz´alez [1] investigated the Hankel convolution satisfying the formula ′ (f♯g)= x−µ−1/2 ′ (f) ′ (g) Hµ Hµ Hµ ′ on new spaces of generalized function, which are subspaces of Hµ (see [1]). Their modified Hankel transform is given by ∞ ( µφ)(t)= √xtJµ(xt)φ(x)dx, t (0, ). (3) H Z0 ∈ ∞ As outlined in the abstract, the current work is to provide a means of defining the Hankel transform hµ,k for any real value of the order in such a way that an −1 inverse Hankel transform hµ,k also exists and is equal to the Hankel transform itself, although Zemanian claimed that it is not true for < 1/2 in [13]. It is − the property of the inverse transform that makes this extension of the Hankel transform significant, since it serves well in constructing the Hankel convolution of arbitrary order, which has never be studied in the past as far as we know.

2. The Hankel Transform of Arbitrary Order

In order to extend the Hankel transform in (3) to generalized functions, Zema- nian [13] defined the following testing space Hµ.

Definition 2.1. For any real number , a function φ(x) is in Hµ if and only if it is complex-valued and smooth on R+, and for each pair of nonnegative integers m and k, µ m −1 k −µ−1/2 γm,k(φ)= sup x (x D) x φ(x) < . x∈R+ ∞ µ Obviously, Hµ is a linear space. Also, each γm,k is a seminorm on Hµ, and the collection γµ ∞ is a multinorm because the γµ are norms. The { m,k}m,k=0 m,0 250 C.K. Li topology of H is that generated by γµ ∞ . µ { m,k}m,k=0 1/2 Clearly, the mapping φ(x) x φ(x) is an isomorphism from H 1 into → µ− 2 Hµ. It follows from equations (1) and (3) that (h φ)(t)= t−1/2 (x1/2φ)(t). (4) µ Hµ Using the fact that is an automorphism on H for 1/2 (see [13]), we Hµ µ ≥− obtain Theorem 2.1. For > 1/2, the Hankel transform h is an automorphism − µ on H 1 . µ− 2 We now define two modified differential operators Nµ and Mµ, and a linear −1 integral operator Nµ by −1 µ −µ Nµφ(x) = (D x )φ(x)= x Dx φ(x), − −1 −1 −µ−1 µ+1 Mµφ(x) = (D + x + x )φ(x)= x Dx φ(x), x −1 µ −µ Nµ φ(x)= x t φ(t)dt, Z∞ which will be used to prove the inverse property h = h−1 for R. µ,k µ,k ∈ Applying identity (4) and the Zemanian’s results for in [13], we can Hµ easily list the following.

Lemma 2.1. Nµ is a continuous linear mapping from H 1 into H 1 µ− 2 µ+ 2 −1 while N is a continuous linear mapping of H 1 into H 1 . µ µ+ 2 µ− 2 Lemma 2.2. The mapping φ Mµφ is linear and continuous from H 1 → µ+ 2 into H 1 . Furthermore, for any integer n and for any , the mapping φ(x) µ− 2 n → x φ(x) is an isomorphism from H 1 into H 1 . µ− 2 µ− 2 +n Lemma 2.3. Let > 1/2. If φ H 1 , then − ∈ µ− 2 h ( tφ)= N h φ, (5) µ+1 − µ µ h (N φ)= xh φ, (6) µ+1 µ − µ h ( t2φ)= M N h φ, (7) µ − µ µ µ h (M N φ)= x2h φ, (8) µ µ µ − µ hµ(tφ)= Mµhµ+1φ, if φ H 1 , (9) ∈ µ+ 2 hµ(Mµφ)= xhµ+1φ, if φ H 1 . (10) ∈ µ+ 2

Let R and a positive integer k such that + k > 1 . Assume that ∈ − 2 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 251

φ H 1 . Define the Hankel transform of arbitrary order hµ,k on H 1 by ∈ µ− 2 µ− 2 k −k Φ(x)= hµ,k(φ(y)) = ( 1) x hµ+kNµ+k−1 Nµ+1Nµφ(y), − −1 and let Φ(x) H 1 and define an inverse transform h on H 1 by ∈ µ− 2 µ,k µ− 2 φ(y)= h−1 (Φ(x)) = ( 1)kN −1N −1 N −1 h xkΦ(x). µ,k − µ µ+1 µ+k−1 µ+k From identity (4) and the Zemanian’s Lemma (on p. 164 of [13]), we have the following theorem.

Theorem 2.2. hµ, k is an automorphism on H 1 for R. Its inverse is µ− 2 ∈ h−1 , and h = h if > 1 . µ, k µ, k µ − 2 Note that the definition of hµ,k is independent of the choice of k so long as k + > 1 . Indeed if k>p> 1 , then h = h by Theorem 2.2, − 2 − − 2 µ+p,k−p µ+p hence h φ = ( 1)kx−kh N ...N φ µ, k − µ+k µ+k−1 µ = ( 1)px−p( 1)k−px−(k−p)h N ...N N ...N φ − − µ+p+k−p µ+p+k−p−1 µ+p µ+p−1 µ = ( 1)px−ph N ...N φ = ( 1)px−ph N ...N φ = h φ. − µ+p,k−p µ+p−1 µ − µ+p µ+p−1 µ µ,p Zemanian claimed in [13] (on p. 165) that = −1 (which implies Hu, k Hu, k h = h−1 , since (h φ)(t)= t−1/2 (x1/2φ)(t)) when < 1/2. However, u, k u, k µ,k Hµ,k − he did not give any counterexample. F.H. Kerr [6] introduced complex fractional powers of Hankel transforms α on H to show that = −1 . In the present Hµ µ Hµ,k Hµ,k work, we are able to give a direct and simple proof that h = h−1 for R u, k u, k ∈ with the help of the following identity in [13] D x−µJ (xy)= yx−µJ (xy). (11) x µ − µ+1 This inverse property will play an important role in defining the Hankel convo- lutions in Section 3.

Lemma 2.4. Nµhµ, k(φ)= hµ+1,k( yφ) for φ H 1 . − ∈ µ− 2 Proof. By definition k −k hu, kφ = ( 1) x hµ+kNµ+k−1 Nµ+1Nµφ(y) − k −k µ+k −1 k −µ = ( 1) x hµ+ky (y D) y φ(y) − ∞ k −k µ+k −1 k −µ = ( 1) x yJµ+k(xy)y (y D) y φ(y)dy. − Z0 It follows that ∞ k −k µ+k −1 k −µ Nµhu, k(φ) = ( 1) Nµx yJµ+k(xy)y (y D) y φ(y)dy. − Z0 252 C.K. Li

By equation (11), we have −k µ −µ−k Nµx Jµ+k(xy) = x Dx Jµ+k(xy) = yx−kJ (xy). − µ+1+k Hence, ∞ k −k µ+1+k −1 k −µ−1 Nµhu, k(φ) = ( 1) x yJµ+1+k(xy)y (y D) y ( yφ(y))dy − Z0 − = h ( yφ).  µ+1,k − Theorem 2.3. Let be any fixed real number and let k be any positive −1 integer such that + k > 1/2. Then hµ, k = h on H 1 . − µ, k µ− 2 Proof. By Lemma 2.4 we have N h (φ)= h ( yφ). µ µ, k µ+1,k − Applying Nµ+1 to both sides, we obtain N N h (φ)= N h ( yφ)= h ( 1)2y2φ . µ+1 µ µ, k µ+1 µ+1, k − µ+2,k{ − } Repeating this process, we get N N N h (φ)= h ( 1)kykφ . µ+k−1 µ+1 µ µ, k µ+k, k{ − } Therefore, N N N h (φ) = ( 1)kh (ykφ) µ+k−1 µ+1 µ µ, k − µ+k and we finally come to h (φ) = ( 1)kN −1N −1 N −1 h ykφ(y)= h−1 (φ). µ, k − µ µ+1 µ+k−1 µ+k µ, k This completes the proof of Theorem 2.3.

3. The Hankel Convolution of Arbitrary Order

According to the author’s knowledge, the Hankel convolution of arbitrary order (in particular, 1/2) has not been investigated so far on any spaces of ≤ − interest. In this section, we use the inverse property to construct the Hankel convolution of arbitrary order on the spaces (H 1 ,H 1 ) and (Sµ,H 1 ) µ− 2 µ− 2 µ− 2 respectively. Let R and any positive integer k such that + k > 1 . Assume that ∈ − 2 φ and ψ are in H 1 . Define the Hankel convolution of arbitrary order h(x) of µ− 2 φ and ψ by ∞ ∞ ∞ k −k µ+k+1 −1 k −µ µ+k+1 −1 k −µ h(x) = ( 1) x y (y Dy) y φ(y)u (u Du) u − Z0 Z0 Z0 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 253

ψ(u)tµ+k+1(t−1D )k t−2µ−2kJ (yt)J (ut) J (xt)dudydt. (12) t { µ+k µ+k } µ+k It is obvious to see that this convolution is commutative and we are going to prove the following theorem. Theorem 3.1. Let R and any positive integer k such that +k > 1 . ∈ − 2 Then the function h(x) in (12) exists and the exchange formula −µ (hµ,kh)(t)= t (hµ,kφ)(t)(hµ,kψ)(t) holds on (H 1 ,H 1 ). µ− 2 µ− 2 Proof. Clearly, ∞ ∞ k −k µ+k+1 −1 k −µ µ+k+1 h(x) = ( 1) x hµ+k y (y Dy) y φ(y)u − Z0 Z0 (u−1D )ku−µψ(u)tµ+k(t−1D )k t−2µ−2kJ (yt)J (ut) dudy. u t { µ+k µ+k } The following integral, for every k ∞ ∞ µ+k+1 −1 k −µ µ+k+1 −1 k −µ y (y Dy) y φ(y)u (u Du) u ψ(u) Z0 Z0 (t−1D )k t−2µ−2kJ (yt)J (ut) dudy t { µ+k µ+k } is uniformly convergent on every compact subset of R+ with respect to t since φ and ψ are in H 1 . Therefore, µ− 2 h(x) = ( 1)kx−kh N N (t−µ−2kh (yµ+k(y−1D )ky−µφ(y)) − µ+k µ+k−1 µ µ+k y µ+k −1 k −µ −µ hµ+k(u (u Du) u ψ(y))) = hµ,k(t hµ,kφhµ,kψ). −1 The result follows immediately by hµ,k = hµ,k due to Theorem 2.3. This com- pletes the proof of Theorem 3.1. In particular, we get for > 1/2 − ∞ −µ −µ h(x)= hµ(t hµφ hµψ)= tJµ(xt)t hµφ hµψdt Z0 ∞ ∞ ∞ 1−µ = t Jµ(xt) uvJµ(ut)Jµ(vt)φ(u)ψ(v)dudvdt Z0 Z0 Z0 ∞ ∞ ∞ 1−µ = uvφ(u)ψ(v)dudv t Jµ(xt)Jµ(ut)Jµ(vt)dt. Z0 Z0 Z0 This procedure is permissible due to the Fubini Theorem. Furthermore, it is well known (see [10]) that ∞ 1−µ t Jµ(xt)Jµ(ut)Jµ(vt)dt Z0 254 C.K. Li

21−3µ = (xuv)−µ[x2 (u v)2]µ−1/2[(u + v)2 x2]µ−1/2 √πΓ( + 1/2) − − + − + where s(x) if s(x) 0, s (x)= ≥ +  0 if s(x) < 0. Therefore, 21−3µx−µ h(x)= [x2 (u v)2]µ−1/2 √πΓ( + 1/2)ZZ u+v>x, |u−v| 1/2 with the normal addition and scalar multi- µ − plication as ∞ + µ+2k+1 Sµ = φ C(R ) y φ(y) dy < and k = 0, 1, .  ∈ | Z0 | | ∞  Clearly H 1 is a proper subset of Sµ according to Lemma 5.2.1 in [13] and we µ− 2 have the following theorem. −µ Theorem 3.2. The function t (hµφ)(t) is a multiplier of H 1 for any µ− 2 φ S . ∈ µ Proof. Obviously, ∞ −µ −µ t (hµφ)(t)= yt Jµ(yt)φ(y)dy Z0 and (t−1D)kt−µJ (yt) = ( y)kt−µ−kJ (yt). µ − µ+k The integral ∞ ∞ −1 k −µ µ+k+1 k Jµ+k(yt) y(t D) t Jµ(yt)φ(y)dy = y ( y) µ+k φ(y)dy Z0 Z0 − (yt) µ+k is uniformly convergent with respect to t since the term Jµ+k(yt)/(yt) is bounded on 0

Acknowledgements

The author would like to thank Dr. J.J. Betancor and Dr. L.E. Britvina for their very constructive suggestions and corrections in several places. The author is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

References

[1] J.J. Betancor, B.J. Gonz´alez, A convolution operation for a distributional Hankel transformation, Stud. Math., 117 (1995), 57-72.

[2] L.E. Britvina, Generalized convolutions for the Hankel transform and re- lated integral operators, Math. Nachr., 280 (2007), 962-970.

′ [3] B.J. Gonz´alez, E.R. Negrin, Convolution over the spaces Sk, J. Math. Anal. Appl., 190 (1995), 829-843.

′ [4] B.J. Gonz´alez, E.R. Negrin, Fourier transform over the spaces Sk, J. Math. Anal. Appl., 194 (1995), 780-798.

[5] I.I. Hirschman, Jr., Variation diminishing Hankel transforms, J. Analyse Math., 8 (1960/61), 307-336.

[6] F.H. Kerr, Fractional powers of Hankel transforms in the Zemanian spaces, J. Math. Anal. and Appl., 166 (1992), 65-83.

[7] E.L. Koh, C.K. Li, The Hankel transformation of Banach-space-valued generalized functions, Proc. Amer. Math. Soc., 119 (1993), 153-163.

[8] E.L. Koh, C.K. Li, The kernel theorem on the space [H A; B], Proc. µ × Amer. Math. Soc., 123 (1995), 177-182.

[9] C.K. Li, A kernel theorem from the Hankel transform in Banach spaces, . Spec. Funct., 16 (2005), 565-681.

[10] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integral and Series, Vol- ume 2, Special Functions, Gordon and Breach (1986).

[11] J. de Sousa Pinto, A generalized Hankel convolution, SIAM J. Math. Anal., 16 (1985), 1335-1346. 256 C.K. Li

[12] V.K. Tuan, M. Saigo, Convolution of Hankel transform and its application to an integral involving Bessel functions of first kind, Internat. J. Math. and Math. Sci., 3 (1995), 545-550.

[13] A.H. Zemanian, Generalized Integral Transformations, Interscience, New York (1965).

[14] Ya.I. Zhitomirskii, The Cauchy problem for systems of linear partial dif- ferential equations with differential operators of Bessel type, II, Mat. Sb., Math. USSR-Sb., 36 (1955), 299-310.