On self-reciprocal functions under a class of integral transforms

We use the fact that a rather general class of integral transforms—are equivalent to hyperdifferential operators, to formulate the problem of self-reciprocal functions under these transforms as an eigenvalue problem for (second-order) differential operators. We thus find the solution for Fourier, Hankel, bilateral Lapace, Bargmann, Weierstrass-Gauss and Barut-Girardello transforms. These involve the Schrödinger attractive and repulsive harmonic oscillator and/or centrifugal potentials wavefunctions. A general concept of ’’self-reproducing’’ functions is introduced which includes all of the above plus linear potential wavefunctions. In particular, two new generalized bases for Bargmann’s Hilbert space of analytic functions are found.