  While exploiting the generalized Parseval equality for the Mellin transform,
we derive the reciprocal inverse operator in the weighted L_2-space related to
the Hilbert transform on the nonnegative half-axis. Moreover, employing the
convolution method, which is based on the Mellin-Barnes integrals, we prove the
corresponding convolution and Titchmarsh's theorems for the half-Hilbert
transform. Some applications to the solvability of a new class of singular
integral equations are demonstrated. Our technique does not require the use of
methods of the Riemann-Hilbert boundary value problems for analytic functions.
The same approach will be applied in the forthcoming research to invert the
half-Hartley transform and to establish its convolution theorem.

Classical Analysis and ODEs