  The method of realizing certain self-reciprocal transforms as (absolute)
scattering, previously presented in summarized form in the case of the Fourier
cosine and sine transforms, is here applied to the self-reciprocal transform
f(y)-> H(f)(x) = \int_0^\infty J_0(2\sqrt{xy})f(y) dy, which is isometrically
equivalent to the Hankel transform of order zero and is related to the
functional equations of the Dedekind zeta functions of imaginary quadratic
fields. This also allows to re-prove and to extend theorems of de Branges and
V. Rovnyak regarding square integrable functions which are self-or-skew
reciprocal under the Hankel transform of order zero. Related integral formulae
involving various Bessel functions are all established internally to the
method. Fredholm determinants of the kernel J_0(2\sqrt{xy}) restricted to
finite intervals (0,a) give the coefficients of first and second order
differential equations whose associated scattering is (isometrically) the
self-reciprocal transform H, closely related to the function
Gamma(1-s)/Gamma(s). Remarkable distributions involved in this analysis are
seen to have most natural expressions as (difference of) boundary values (i.e.
hyperfunctions.) The present work is completely independent from the previous
study by the author on the same transform H, which centered around the
Klein-Gordon equation and relativistic causality. In an appendix, we make a
simple-minded observation regarding the resolvent of the Dirichlet kernel as a
Hilbert space reproducing kernel.
