zbMATH — the first resource for mathematics

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.

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