International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 66, Pages 3603-3616
doi:10.1155/S0161171204301365
On the Lebedev transformation in Hardy's spaces
Department of Pure Mathematics, Faculty of Science, University of Porto, 687 Campo Alegre Street, Porto 4169-007, Portugal
Received 31 January 2003
Copyright © 2004 Semyon B. Yakubovich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.