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 H2,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 Ĥ2,A ≡ Ĥ2((−A,A);|Γ(1+Rez+iτ)|2dτ), 0 < A < 1, of analytic functions f(z),z = Rez+ iτ , in the strip |Rez| ≤ A. Boundedness and inversion properties of the Lebedev transformation from this space into the space L2(R+;x−1dx) are considered. When Rez = 0, we derive the familiar Plancherel theorem for the Kontorovich-Lebedev transform.