International Journal of Pure and Applied Mathematics Volume 55 No ,

Transcription

1 International Journal of Pure and Applied Mathematics Volume 55 No. 9, THE HANKEL CONVOLUTION OF ARBITRARY ORDER C.K. Li Department of Mathematics and Computer Science Brandon University Brandon, Manitoba, R7A 6A9, CANADA Abstract: Let µ > /. The classical Hankel transform is defined by (h µ φ)(t) = xj µ (xt)φ(x)dx t (, ), () where J µ (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 h µ,k based on the two differential operators and show that h µ,k = h µ,k on H µ for µ R. Furthermore, the Hankel convolution of arbitrary order is introduced with the following identity on the spaces (H µ (h µ,k h)(t) = t µ (h µ,k φ)(t)(h µ,k ψ)(t), H µ ) and (S µ, H µ ) respectively. AMS Subject Classification: 46F Key Words: Hankel transform, Bessel function, Hankel convolution and automorphism. Introduction Let R + = (, ) and a weight function ν(x) > in R +. We define a set of functions L(R +,ν(x)) as L(R +,ν(x)) = {f(x) f(x) ν(x)dx < }. 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, 9 c 9 Academic Publications

2 48 C.K. Li [5] and []) 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 995, Tuan and Saigo [] used the following commutative Hankel convolution h(x) of a function f(x) with a function g(x) (due to Zhitomirskii, see [4]), h(x) = 3µ x µ πγ(µ + /) [x (u v) ] µ / u+v>x, u v <x [(u + v) x ] µ / (uv) µ f(u)g(v)dudv x (, ), () and proved Theorem. using the Hankel transform in (). Theorem.. Let Reµ > / and f(x), g(x) L(R +, x). Then the function h(x) of () exists and the convolutional identity holds. (h µ h)(t) = t µ (h µ f)(t)(h µ g)(t) Recently, Britvina studied some polyconvolutions of the Hankel transform in [], where she defined the following functions h (t) and h (t), and obtained Theorem. and Theorem.3, t+v h (t) = t µ dv u µ+ f(u)g(v)p (t;u,v)du where t µ dv h (t) = t µ t µ du t v t+v t v <t<u+v t u P j (t;u,v) = v µ P / µ π ν / (cos s j)sin µ / s j, Q j (t;u,v) = sin[(ν µ)π]exp π3/ j u µ+ f(u)g(v)q (t;u,v)du, u µ f(u)g(v)p (t;u,v)dudv u µ f(u)g(v)q (t;u,v)dv, { (µ )π } v µ Q / µ ν / (chr j)sh µ / r j, j =,, and P µ ν, Q µ ν are the associated Legendre functions of the first and the second kind, respectively, and tv cos s = t + v u, tvchr = u t v, uv cos s = u + v t, uvchr = t u v. Theorem.. Suppose that f(t), g(t) L(R +, t) and Reµ > /,

3 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 49 Reν > Reµ. Then the function h (t) exists and the relation holds. (h ν h )(x) = x ν (h µ f)(x)(h ν g)(x) Theorem.3. Assume that f(t), g(t) L(R +, t) and Reµ > /, Reν > (Re µ 3)/4. Then the function h (t) exists and the following relation is satisfied. (h µ h )(x) = x µ (h ν f)(x)(h ν g)(x) Inspired by the studies of González and Negrin ([3] and [4]) on the convolution and Fourier transform, Betancor and González [] investigated the Hankel convolution satisfying the formula H µ (f g) = x µ / H µ (f)h µ (g) on new spaces of generalized function, which are subspaces of H µ (see []). Their modified Hankel transform is given by (H µ φ)(t) = xtjµ (xt)φ(x)dx, t (, ). (3) As outlined in the abstract, the current work is to provide a means of defining the Hankel transform h µ,k for any real value of the order µ in such a way that an inverse Hankel transform h µ,k also exists and is equal to the Hankel transform itself, although Zemanian claimed that it is not true for µ < / in [3]. 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.. The Hankel Transform of Arbitrary Order In order to extend the Hankel transform in (3) to generalized functions, Zemanian [3] defined the following testing space H µ. Definition.. For any real number µ, a function φ(x) is in H µ if and only if it is complex-valued and smooth on R +, and for each pair of nonnegative integers m and k, γ µ m,k (φ) = sup x R + x m (x D) k x µ / φ(x) <. Obviously, H µ is a linear space. Also, each γ µ m,k is a seminorm on H µ, and the collection {γ µ m,k } m,k= is a multinorm because the γµ m, are norms. The

4 5 C.K. Li topology of H µ is that generated by {γ µ m,k } m,k=. Clearly, the mapping φ(x) x / φ(x) is an isomorphism from H µ H µ. It follows from equations () and (3) that into (h µ φ)(t) = t / H µ (x / φ)(t). (4) Using the fact that H µ is an automorphism on H µ for µ / (see [3]), we obtain Theorem.. For µ > /, the Hankel transform h µ is an automorphism. on H µ We now define two modified differential operators N µ and M µ, and a linear integral operator N µ by N µ φ(x) = (D µx )φ(x) = x µ Dx µ φ(x), M µ φ(x) = (D + µx + x )φ(x) = x µ Dx µ+ φ(x), x Nµ φ(x) = xµ t µ φ(t)dt, which will be used to prove the inverse property h µ,k = h µ,k for µ R. Applying identity (4) and the Zemanian s results for H µ in [3], we can easily list the following. Lemma.. N µ is a continuous linear mapping from H µ while N µ is a continuous linear mapping of H µ+ into H µ. into H µ+ Lemma.. The mapping φ M µ φ is linear and continuous from H µ+. Furthermore, for any integer n and for any µ, the mapping φ(x) into H µ x n φ(x) is an isomorphism from H µ into H µ Lemma.3. Let µ > /. If φ H µ +n., then h µ+ ( tφ) = N µ h µ φ, (5) h µ+ (N µ φ) = xh µ φ, (6) h µ ( t φ) = M µ N µ h µ φ, (7) h µ (M µ N µ φ) = x h µ φ, (8) h µ (tφ) = M µ h µ+ φ, if φ H µ+, (9) h µ (M µ φ) = xh µ+ φ, if φ H µ+. () Let µ R and a positive integer k such that µ + k >. Assume that

5 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 5 φ H µ. Define the Hankel transform of arbitrary order h µ,k on H µ Φ(x) = h µ,k (φ(y)) = ( ) k x k h µ+k N µ+k N µ+ N µ φ(y), and let Φ(x) H µ and define an inverse transform h µ,k on H µ φ(y) = h µ,k (Φ(x)) = ( )k N µ N µ+ N µ+k h µ+kx k Φ(x). From identity (4) and the Zemanian s Lemma (on p. 64 of [3]), we have the following theorem. Theorem.. h µ, k is an automorphism on H µ for µ R. Its inverse is h µ, k, and h µ, k = h µ if µ >. Note that the definition of h µ,k is independent of the choice of k so long as k + µ >. Indeed if k > p > µ, then h µ+p,k p = h µ+p by Theorem., hence h µ, k φ = ( ) k x k h µ+k N µ+k...n µ φ = ( ) p x p ( ) k p x (k p) h µ+p+k p N µ+p+k p...n µ+p N µ+p...n µ φ = ( ) p x p h µ+p,k p N µ+p...n µ φ = ( ) p x p h µ+p N µ+p... N µ φ = h µ,p φ. Zemanian claimed in [3] (on p. 65) that H u, k H u, k (which implies h u, k h u, k, since (h µ,kφ)(t) = t / H µ,k (x / φ)(t)) when µ < /. However, he did not give any counterexample. F.H. Kerr [6] introduced complex fractional powers of Hankel transforms Hµ α on H µ to show that H µ,k = H µ,k. In the present work, we are able to give a direct and simple proof that h u, k = h u, k for µ R with the help of the following identity in [3] D x x µ J µ (xy) = yx µ J µ+ (xy). () This inverse property will play an important role in defining the Hankel convolutions in Section 3. Lemma.4. N µ h µ, k (φ) = h µ+, k ( yφ) for φ H µ. Proof. By definition It follows that h u, k φ = ( ) k x k h µ+k N µ+k N µ+ N µ φ(y) = ( ) k x k h µ+k y µ+k (y D) k y µ φ(y) = ( ) k x k yj µ+k (xy)y µ+k (y D) k y µ φ(y)dy. N µ h u, k (φ) = ( ) k N µ x k yj µ+k (xy)y µ+k (y D) k y µ φ(y)dy. by by

6 5 C.K. Li By equation (), we have Hence, N µ x k J µ+k (xy) = x µ Dx µ k J µ+k (xy) = yx k J µ++k (xy). N µ h u, k (φ) = ( ) k x k yj µ++k (xy)y µ++k (y D) k y µ ( yφ(y))dy = h µ+, k ( yφ). Theorem.3. Let µ be any fixed real number and let k be any positive integer such that µ + k > /. Then h µ, k = h µ, k on H µ. Proof. By Lemma.4 we have N µ h µ, k (φ) = h µ+, k ( yφ). Applying N µ+ to both sides, we obtain N µ+ N µ h µ, k (φ) = N µ+ h µ+, k ( yφ) = h µ+, k {( ) y φ}. Repeating this process, we get Therefore, and we finally come to N µ+k N µ+ N µ h µ, k (φ) = h µ+k, k {( ) k y k φ}. N µ+k N µ+ N µ h µ, k (φ) = ( ) k h µ+k (y k φ) h µ, k (φ) = ( ) k N µ N µ+ N µ+k h µ+ky k φ(y) = h µ, k (φ). This completes the proof of Theorem The Hankel Convolution of Arbitrary Order According to the author s knowledge, the Hankel convolution of arbitrary order (in particular, µ /) 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 µ respectively., H µ ) and (S µ, H µ Let µ R and any positive integer k such that µ + k >. Assume that φ and ψ are in H µ. Define the Hankel convolution of arbitrary order h(x) of φ and ψ by h(x) = ( ) k x k y µ+k+ (y D y ) k y µ φ(y)u µ+k+ (u D u ) k u µ )

7 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 53 ψ(u)t µ+k+ (t D t ) k {t µ k J µ+k (yt)j µ+k (ut)}j µ+k (xt)dudydt. () It is obvious to see that this convolution is commutative and we are going to prove the following theorem. Theorem 3.. Let µ R and any positive integer k such that µ+k >. Then the function h(x) in () exists and the exchange formula holds on (H µ, H µ ). Proof. Clearly, h(x) = ( ) k x k h µ+k (h µ,k h)(t) = t µ (h µ,k φ)(t)(h µ,k ψ)(t) y µ+k+ (y D y ) k y µ φ(y)u µ+k+ (u D u ) k u µ ψ(u)t µ+k (t D t ) k {t µ k J µ+k (yt)j µ+k (ut)}dudy. The following integral, for every k y µ+k+ (y D y ) k y µ φ(y)u µ+k+ (u D u ) k u µ ψ(u) (t D t ) k {t µ k J µ+k (yt)j µ+k (ut)}dudy is uniformly convergent on every compact subset of R + with respect to t since φ and ψ are in H µ. Therefore, h(x) = ( ) k x k h µ+k N µ+k N µ (t µ k h µ+k (y µ+k (y D y ) k y µ φ(y)) h µ+k (u µ+k (u D u ) k u µ ψ(y))) = h µ,k (t µ h µ,k φh µ,k ψ). The result follows immediately by h µ,k = h µ,k due to Theorem.3. This completes the proof of Theorem 3.. In particular, we get for µ > / h(x) = h µ (t µ h µ φh µ ψ) = = t µ J µ (xt) = tj µ (xt)t µ h µ φh µ ψdt uvφ(u)ψ(v)dudv uvj µ (ut)j µ (vt)φ(u)ψ(v)dudvdt t µ J µ (xt)j µ (ut)j µ (vt)dt. This procedure is permissible due to the Fubini Theorem. Furthermore, it is well known (see []) that t µ J µ (xt)j µ (ut)j µ (vt)dt

8 54 C.K. Li where Therefore, = 3µ (xuv) µ [x (u v) ] µ / πγ(µ + /) + [(u + v) x ] µ / + s + (x) = h(x) = 3µ x µ πγ(µ + /) { s(x) if s(x), if s(x) <. [x (u v) ] µ / u+v>x, u v <x which coincides with equation () in the introduction. [(u + v) x ] µ / (uv) µ φ(u)ψ(v)dudv In order to extend the Hankel convolution to a larger function space, we define the space S µ for µ > / with the normal addition and scalar multiplication as S µ = { φ C(R + ) } y µ+k+ φ(y) dy < and k =,,. Clearly H µ is a proper subset of S µ according to Lemma 5.. in [3] and we have the following theorem. Theorem 3.. The function t µ (h µ φ)(t) is a multiplier of H µ φ S µ. and Proof. Obviously, The integral t µ (h µ φ)(t) = yt µ J µ (yt)φ(y)dy (t D) k t µ J µ (yt) = ( y) k t µ k J µ+k (yt). y(t D) k t µ J µ (yt)φ(y)dy = y µ+k+ ( y) k J µ+k(yt) φ(y)dy (yt) µ+k for any is uniformly convergent with respect to t since the term J µ+k (yt)/(yt) µ+k is bounded on < yt < and φ S µ. This implies that (t D) k t µ (h µ φ)(t) is bounded and thus a multiplier of H µ. Theorems 3. and. allow us directly define the Hankel convolution h(x) in H µ as h(x) = h µ (t µ h µ φh µ ψ) where (φ, ψ) (S µ, H µ ).

9 THE HANKEL CONVOLUTION OF ARBITRARY ORDER 55 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 [] J.J. Betancor, B.J. González, A convolution operation for a distributional Hankel transformation, Stud. Math., 7 (995), [] L.E. Britvina, Generalized convolutions for the Hankel transform and related integral operators, Math. Nachr., 8 (7), [3] B.J. González, E.R. Negrin, Convolution over the spaces S k, J. Math. Anal. Appl., 9 (995), [4] B.J. González, E.R. Negrin, Fourier transform over the spaces S k, J. Math. Anal. Appl., 94 (995), [5] I.I. Hirschman, Jr., Variation diminishing Hankel transforms, J. Analyse Math., 8 (96/6), [6] F.H. Kerr, Fractional powers of Hankel transforms in the Zemanian spaces, J. Math. Anal. and Appl., 66 (99), [7] E.L. Koh, C.K. Li, The Hankel transformation of Banach-space-valued generalized functions, Proc. Amer. Math. Soc., 9 (993), [8] E.L. Koh, C.K. Li, The kernel theorem on the space [H µ A;B], Proc. Amer. Math. Soc., 3 (995), [9] C.K. Li, A kernel theorem from the Hankel transform in Banach spaces, Integral Transform. Spec. Funct., 6 (5), [] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integral and Series, Volume, Special Functions, Gordon and Breach (986). [] J. de Sousa Pinto, A generalized Hankel convolution, SIAM J. Math. Anal., 6 (985),

10 56 C.K. Li [] 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 (995), [3] A.H. Zemanian, Generalized Integral Transformations, Interscience, New York (965). [4] Ya.I. Zhitomirskii, The Cauchy problem for systems of linear partial differential equations with differential operators of Bessel type, II, Mat. Sb., Math. USSR-Sb., 36 (955), 99-3.