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In this article, we prove several formulas about the interchange between the product and the convolution of the n-dimensional Hankel transforms. In fact, we obtain the following formulas: 1. 2. where D is the constant given by 3. where C is the constant given by 4. where D is the constant given by 5. δ( k )(u(|x|2)) * δ(l)(u(|x|2)) = D δ( k +l+( n −2)/2)(u(|x|2)), here D is the constant given by 6.

Let $f(z, \lambda), z\in {\open C}$ , be an entire function of the variables $z, \lambda, f(z, \lambda) = \sum_{\nu = 0}^\infty a_\nu(\lambda)z^\nu$ . Let us consider the family of distributions of the form [1], p. 285, $T(P\pm io, \lambda) = (P \pm io)^\lambda \quad f(P\pm io, \lambda) = (P\pm io)^\lambda \sum_{\nu=0}^\infty a_\nu(\lambda)(P\pm io)^\lambda$ In this paper we generalize the concept of the classic one-dimensional Laplace transform to certain distributions ( $P\pm io)^\lambda$ (an important contribution of Gelfand, cf. [1], p. 274). We define the causal, anticausal distributions R_\alpha (P\pm io,n)$ (cf. [3.1], p. 4). This distribution is an elementary solution of the homogeneous ultrahyperbolic operator iterated k -times. We observe that the distributional function $R_\alpha(P\pm io,n)$ is a (causal, anticausal) analogue of the kernel due to A. P. Caldero n, Aronszjan-Smith and L. Schwartz (cf. M.I.T. [1958], Ann. Inst. Fourier [1961] and Hermann, Paris [1960], respectively).

In this paper, we consider the solution of the equation \diamondsuit^{k}(P\pm i0) = \sum_{r=0}^{m} C_{r}\diamondsuit^{r}\delta where \diamondsuit^{k} is introduced and named as the Diamond operator iterated k -times and is defined by \diamondsuit^{k} = \left[\left({\partial^{2} \over \partial x_{1}^{2}}+ \cdots + {\partial^{2} \over \partial x_{p}^{2}}\right)^{2} - \left({\partial^{2} \over \partial x_{p+1}^{2}} + \cdots + {\partial^{2} \over \partial x_{p+q}^{2}}\right)^{2}\right]^{k}. Let x = (x_{1},x_{2},\ldots,x_{n}) be a point of the n -dimensional Euclidean space. Consider a non-degenerate quadratic form in n variables of the form P = P(x) = x_{1}^{2} + \cdots + x_{p}^{2} - x_{p+1}^{2} - \cdots - x_{p+q}^{2} , where p + q = n , C_{r} is a constant, \delta is the delta distribution \diamondsuit^{0}\delta =\delta and k = 0,1,\ldots. The distributions (P\pm i0)^{\lambda} are defined by (P\pm i0)^{\lambda} = \lim_{\varepsilon \rightarrow 0}\ \{P\pm i\varepsilon \vert x\vert^{2}\}^{\lambda} , where \vare...

In this note we obtain the following products of the n -dimensional distributions: (i) (m^{2} + {\bi P} \pm io)^{\lambda} \cdot (m^{2} + {\bi P} \pm io)^{\mu} = (m^{2} + {\bi P} \pm io)^{\lambda + \mu} (cf. (1.10)) (ii) ({\bi P} \pm io)^{\lambda} \cdot ({\bi P} \pm io)^{\mu} = ({\bi P} \pm io)^{\lambda + \mu} (cf. (1.11)) (iii) (m^{2} + {\bi P})_{+}^{\lambda} \cdot (m^{2} + {\bi P})_{+}^{\mu} = (m^{2} + {\bi P})_{+}^{\lambda + \mu} (cf. (3.4)) (iv) (m^{2} + {\bi P})_{-}^{\lambda} \cdot (m^{2} + {\bi P})_{-}^{\mu} = (m^{2} + {\bi P})_{-}^{\lambda + \mu} (cf. (3.5)) (v) e^{i\pi \lambda}(m^{2} + {\bi P})_{-}^{\lambda} \cdot (m^{2} + {\bi P})_{+}^{\mu} = -e^{i\pi\mu}(m^{2} + {\bi P})_{-}^{\mu} \cdot (m^{2} + {\bi P})_{+}^{\lambda} (cf. (3.7)) (vi) (-1)^{\lambda} {\bi P}_{-}^{\lambda} {\bi P}_{+}^{\lambda} = (-1)(-1)^{\mu} {\bi P}_{-}^{\mu} \cdot {\bi P}_{+}^{\lambda} (cf. (3.8)) (vii) [(m^{2} + {\bi P} \pm io)^{\lambda} \log^{r} (m^{2} + {\bi P} \pm io)] \cdot [(m^{2} + {\b...

In this Note we evaluate the Laplace transform (cf. formula (16)) of the distributions ([1], formulae (6) and (7), p.289). The distributions are the Fourier transforms of the causal (anticausal) distributions ([2], formula (I.4.1), p.321). The distributions are analogues of the causal (anti-causal) ones of the Calderon-Aronszajn-smith radial kernel.

The Riesz kernel where n is the dimension of the space (cf. form. (1)) can be expressed by means of a Riemann-Liouville integral (cf. form. (13)). The very interesting particular case α=2k expressed by form. (15). The kernel Wα(u,m) has analogous properties to the Riesz kernel (cf. [1]). That is,putting p=1 in (1) we obtain the kernel introduced by Riesz (cf. [1], p.17,[2], p.89, [3], p.179 and [4], p.72). Therefore, we know that ,where is the n-dimensional ultrahyperbolic Klein-Gordon operator, (cf. [5], p.9, form. (2.29)); (cf. [5], p.19, form (IV.9)), here is the n-dimensional ultrahyperbolic Klein-Gordon operator iterated K-times W0(u,m)=δ (cf. [5], p.20, form (VI)). OtherwiseWα(u,m) can be expressed as an infinite, linear combination of Rα(u) of different orders (cf. [5], p.15, form. (3.26)) where Rα(u) is the ultrahyperbolic Riesz (cf. [6], p.72) and Wα(u,m)=Rα(u) (cf. [5], p.22, form. (V2.5). We also study the particular case W0(u,m) (see Note).