On the Plancherel theorem for the Olevskii transform (Q2466265)

We deal here with a class of integral transformations with respect to parameters of hypergeometric functions or the index transforms. In particular, we treat the familiar Olevskii transform, which is associated with the Gauss hypergeometric function as a kernel. It involves, in turn, as particular cases index transforms of the Mehler-Fock type which are used in the mathematical theory of elasticity. Boundedness \(L_2\)-properties for the Olevskii transform are investigated. The Plancherel theorem is proved. It shows that the Olevskii transform is an isometric isomorphism between two weighted \(L_2\)-spaces. More examples of such isomorphisms are exhibited for the Mehler-Fock type transforms.