The idea of expanding a function in terms of the solutions of a second-order differential equation was first presented in a paper by Sturm and Liouville in 1836. But they gave a proof which is now not accepted. The first satisfactory proofs were not constructed until early in the twentieth century. These early proofs by Hilbert and others were established only if the function together with its first and second derivatives is continuous. A presentation based on these early papers and attacking the problem by the sole use of the theory of differential equations is given by E.L. Ince in his book "Ordinary Differential Equations".

Kozakiewicz, W. (Supervisor)