The composition theorems of the Plancherel type are established in the
paper for so-called index transforms by means of the classical Planch
erel theorems for tile Fourier and the Hankel transforms. The known an
d new examples of such transforms are considered, namely, the Kontorov
ich-Lebedev transform, the Mehler-Fock transform, the Lebedev-Skalskay
a transform etc.