International Journal of Pure and Applied Mathematics ————————————————————————– Volume 55 No. 2 2009, 247-256 THE HANKEL CONVOLUTION 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 µ> 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 Bessel function of the first kind and order µ. 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 µ 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 convolutions, 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(µ + 1/2) Z Z u+v>x, |u−v|<x − − · [(u + v)2 x2]µ−1/2(uv)1−µf(u)g(v)dudv x (0, ), (2) − ∈ ∞ and proved Theorem 1.1 using the Hankel transform in (1). Theorem 1.1. Let Re µ > 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 Z Z|t−v|<t<u+v ∞ t−u −µ µ t du u f(u)g(v)Q2(t; u, v)dv, − Z0 Z0 where 1 µ 1/2−µ µ−1/2 Pj(t; u, v)= v P (cos sj)sin sj, √2π ν−1/2 √2 (2µ 1)π µ 1/2−µ µ−1/2 Qj(t; u, v)= sin[(ν µ)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 µ > 1/2, ∈ THE HANKEL CONVOLUTION OF ARBITRARY ORDER 249 Re ν > Re µ 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 µ > 1/2, ∈ Re ν > (2 Re µ 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 Fourier transform, 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 µ 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 µ< 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 µ, 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 µ 1/2 (see [13]), we Hµ µ ≥− obtain Theorem 2.1. For µ> 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 µx )φ(x)= x Dx φ(x), − −1 −1 −µ−1 µ+1 Mµφ(x) = (D + µx + 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 µ 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 µ, 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 µ> 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 µ R and a positive integer k such that µ + 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 µ R. Its inverse is µ− 2 ∈ h−1 , and h = h if µ> 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 + µ> 1 . Indeed if k>p> µ 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.