Optimal free parameters in orthonormal approximations

We consider orthonormal expansions where the basis functions are governed by some free parameters. If the basis functions adhere to a certain differential or difference equation, then an expression can be given for a specific enforced convergence rate criterion as well as an upper bound for the quadratic truncation error. This expression is a function of the free parameters and some simple signal measurements. Restrictions on the differential or difference equation that make this possible are given. Minimization of either the upper bound or the enforced convergence criterion as a function of the free parameters yields the same optimal parameters, which are of a simple form. This method is applied to several continuous-time and discrete-time orthonormal expansions that are all related to classical orthogonal polynomials.