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On the generalized Kontorovich-Lebedev transform. (English) Zbl 0733.46019
The author defines a generalized Kontorovich-Lebedev transform by using the method of adjoints. In this way he obtains an extension of previous results which define the K-L transform of a distribution $$f$$ directly as an application of $$f$$ to $$K_{iz}(x)$$. The method allows to prove the existence of generalized functions whose K-L transform is equal to a given entire function $$F(z)$$ bounded as $$z\to 0$$ and satisfying the asymptotic behaviour $$F(z)=O(z^{2r} \exp (-\pi z/2))$$, $$z\to \infty$$. These results are employed to solve a problem for the wave operator in a wedge with zero initial conditions and boundary conditions of mixed type.

##### MSC:
 46F12 Integral transforms in distribution spaces 44A15 Special integral transforms (Legendre, Hilbert, etc.) 35G15 Boundary value problems for linear higher-order PDEs 45N05 Abstract integral equations, integral equations in abstract spaces