On the Lebedev transformation in Hardy's spaces

We establish the inverse Lebedev expansion with respect to parameters and arguments of the modified Bessel functions for an arbitrary function from Hardy's spaceH2,A,A>0. This gives another version of the Fourier-integral-type theorem for the Lebedev transform. The result is generalized for a weighted Hardy spaceH⌢2,A≡H⌢2((−A,A);|Γ(1+Rez+iτ)|2dτ),0<A<1, of analytic functionsf(z),z=Rez+iτ, in the strip|Rez|≤A. Boundedness and inversion properties of the Lebedev transformation from this space into the spaceL2(ℝ+;x−1dx)are considered. WhenRez=0, we derive the familiar Plancherel theorem for the Kontorovich-Lebedev transform.