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 space H 2 , 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 space H ⌢ 2 , A ≡ H ⌢ 2 ( ( − A , A ) ; | Γ ( 1 + Re z + i τ ) | 2 d τ ) , 0 < A < 1 , of analytic functions f ( z ) , z = Re z + i τ , in the strip | Re z | ≤ A . Boundedness and inversion properties of the Lebedev transformation from this space into the space L 2 ( ℠+ ; x − 1 d x ) are considered. When Re z = 0 , we derive the familiar Plancherel theorem for the Kontorovich-Lebedev transform.