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The \(L_{\nu }^{(\rho)}\)- transformation on McBride’s spaces of generalized functions. (English) Zbl 0971.44002
The authors consider a generalized Laplace transformation \(L^{(\rho)}_{\nu } (\nu \in \mathbb C, \rho >0)\), introduced by E. Krätzel [Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Naturw. Reihe 14, 369-381 (1965; Zbl 0168.10204)]. They show that it holds \[ \int ^{\infty }_0 (L^{(\rho)}_{\nu } \varphi) (x) \psi (x) dx = \int ^{\infty }_0 \varphi (x) (L^{(\rho)}_{\nu } \psi)(x) dx \] for \(\varphi \in F_{p,\mu }\), \(\psi \in F_{p',\mu '}\), if some conditions on \(\nu ,\rho ,p,\mu \) are fulfilled; here \(p',\mu '\) are defined by relations \(1/p + 1/p' = 1\), \(\mu - \mu ' = 1/p-1/p'\), and \(F_{p,\mu }\) \((p\geq 1, \mu \in \mathbb C)\) denotes a test function space, introduced by A. C. McBride [Fractional calculus and integral transforms of generalized functions (1979; Zbl 0423.46029)] (which is a subset of \(C^{\infty }(\mathbb R^+)\)). This formula allows to extend the operator \(L^{(\rho)}_{\nu }\) onto the corresponding distribution spaces.
The authors prove also a result on a composition of this extended operator \(L^{(\rho)}_{\nu }\) with a differential operator.

MSC:
44A10 Laplace transform
46F12 Integral transforms in distribution spaces
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