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

Abstract

Consider the Sturm-Liouville boundary-value problem 1. (1) y″ − q( x) y = − t 2 y, −∞ < a ⩽ x ⩽ b < ∞ 2. (2) y( a) cos α + y′( a) sin α = 0 3. (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) = ∝ a b f( x) φ( x, t) dx, where f( x) ϵ L 2( 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.