ON THE LAPLACE TRANSFORM OF DISTRIBUTIONS OF THE FORM {\bi T}({\bi P} \pm {\bi io}, \bilambda)
Titel:
ON THE LAPLACE TRANSFORM OF DISTRIBUTIONS OF THE FORM {\bi T}({\bi P} \pm {\bi io}, \bilambda)
Auteur:
Trione, Susana Elena
Verschenen in:
Integral transforms and special functions
Paginering:
Jaargang 14 (2003) nr. 3 pagina's 257-261
Jaar:
2003-06
Inhoud:
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).
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