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 s how that if continuity conditions for a family of quadratic forms hold uniformly on an index set for the family, generalized signature appro ximation results hold. We then apply these results to randomized splin e type Sturm-Liouville problems and obtain continuity of the nth eigen value for generalized Sturm-Liouville problems under weak hypotheses.