International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 65-68, Pages 3603-3616
doi:10.1155/S0161171204301365

On the Lebedev transformation in Hardy's spaces

Semyon B. Yakubovich

Department of Pure Mathematics, Faculty of Science, University of Porto, 687 Campo Alegre Street, Porto 4169-007, Portugal

Abstract

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 H⌢2,A≡H⌢2((−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.