# zbMATH — the first resource for mathematics

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
Full Text: