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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
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