Complete orthogonal sets of functions in \(\mathcal {L}^{2}\) which span the space arise naturally as solutions of certain second-order linear differential equations under appropriate boundary conditions, commonly referred to as Sturm-Liouville boundary-value problems, after the Swiss mathematician Jacques Sturm (1803–1855) and the French mathematician Joseph Liouville (1809–1882), who studied these problems and the properties of their solutions.