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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 66, Pages 3603-3616

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 Hindawi Publishing Corporation. 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.


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 H2,AH2((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(+;x1dx) are considered. When Rez=0, we derive the familiar Plancherel theorem for the Kontorovich-Lebedev transform.