On the computation of Gramian matrices for refinable bases on the interval
In this paper, we discuss a simple method for the numerical computation of Gramian matrices, that appears within the construction of multiresolution analysis on the interval. The presented approach covers all, the orthogonal, the semiorthogonal and the biorthogonal cases. We consider constructions, which are based on translates of a known pair of compactly supported functions phi, (phi) over tilde inside the interval, and supplemented by boundary functions. Using the given two-scale-coefficients of these boundary functions, we can reduce the problem to a linear system of equations. Furthermore, we discuss conditions providing that these systems are uniquely solvable. In particular, no integrals have to be computed numerically. Finally, using the Schoenberg spline basis on the interval as an example, we show how to apply the method to a well-known problem.
interval biorthogonalization boundary functions schoenberg spline basis gramian matrix