A generalized approximation theory for quadratic forms: Application to randomized spline type Sturm-Liouville problems

An approximation theory for families of quadratic forms is given. We show that if continuity conditions for a family of quadratic forms hold uniformly on an index set for the family, generalized signature approximation results hold. We then apply these results to randomized spline type Sturm-Liouville problems and obtain continuity of thenth eigenvalue for generalized Sturm-Liouville problems under weak hypotheses.