Abstract:
The generalized Hankel transforms are studied in the
first part of this thesis; these include the Watson transforms
as a special case. For the validity of the reciprocal
relations, a necessary and sufficient condition on
the kernel is proved. The proof involves first changing
the variables so that all the relations can be written in
the form of convolutions, and then applying the Fourier-
Plancherel transforms to reduce the transcendental equations
to simple algebraic equations. In the second part
of the thesis, unitary mappings on the Hilbert space of
square-integrable functions are characterized "analytically".
A specialization of the kernels appearing in
these analytic formulas yields the Watson transforms. A
theorem on obtaining new pairs of kernels from two pairs
of known ones is proved.
