On the inversion of integral transforms associated with Sturm-Liouville problems

AbstractConsider the Sturm-Liouville boundary-value problem 1.(1) y″ − q(x) y = −t2y, −∞ < a ⩽ x ⩽ b < ∞2.(2) y(a) cos α + y′(a) sin α = 03.(3) y(b) cos β + y′(b) sin β = 0, where q(x) is continuous on [a, b]. Let φ(x, t) be a solution of either the initial-value problem (1) and (2) or (1) and (3). In this paper we develop two techniques to invert the integral F(t) = ∝abf(x) φ(x, t) dx, where f(x) ϵ L2(a, b); one technique is based on the construction of some biorthogonal sequence of functions and the other is based on Poisson's summation formula