The solutions of the $n$-dimensional Bessel diamond operator and the Fourier--Bessel transf.pdf

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The solutions of the $n$-dimensional Bessel diamond operator and the Fourier--Bessel transf.pdf

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The solutions of the $n$-dimensional Bessel diamond operator and the Fourier--Bessel transf

a r X i v : m a t h / 0 5 0 3 0 9 1 v 1 [ m a t h .A P ] 5 M a r 2 0 0 5 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 114, No. 4, November 2003, pp. 375–387. Printed in India The solutions of the n-dimensional Bessel diamond operator and the Fourier–Bessel transform of their convolution HU?SEYIN YILDIRIM ? , M ZEKI SARIKAYA and SERMIN O?ZTU?RK Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Afyon, Turkey ? Corresponding Author. E-mail: hyildir@aku.edu.tr; sarikaya@aku.edu.tr; ssahin@aku.edu.tr MS received 21 April 2004; revised 8 September 2004 Abstract. In this article, the operator 3 k B is introduced and named as the Bessel diamond operator iterated k times and is defined by 3 k B = [(Bx1 +Bx2 + · · ·+Bxp) 2 ? (Bxp+1 + · · ·+Bxp+q) 2]k, where p + q = n, Bxi = ? 2 ? x 2 i + 2vi xi ? ? xi , where 2vi = 2αi + 1, αi ? 1 2 [8], xi 0, i = 1,2, . . . ,n, k is a non-negative integer and n is the dimension of R+ n . In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator 3 k B is called the Bessel diamond kernel of Riesz. Then, we study the Fourier–Bessel transform of the elementary solution and also the Fourier–Bessel transform of their convolution. Keywords. Diamond operator; tempered distribution; Fourier–Bessel transform. 1. Introduction Gelfand and Shilov [2] have first introduced the elementary solution of the n-dimensional classical diamond operator. Later, Kananthai [3,4,5] has proved the distribution related to the n-dimensional ultra-hyperbolic equation, the solutions of n-dimensional classical diamond operator and Fourier transformation of the diamond kernel of Marcel Riesz. Furthermore, Kananthai [4] has showed that the solution of the convolution form u(x) = (?1)kS2k(x)?R2k(x) is a unique elementary solution of the 3 k u(x) = δ . In this article, we will define the Bessel ultra-hyperbolic type operator iterated k times with x ∈ R+ n = {x: x = (x1, . . . ,xn),x