Computing connection coefficients of compactly supported wavelets on bounded intervals

Daubechies wavelet basis functions have many properties that make them desirable as a basis for a Galerkin approach to solving PDEs: they are orthogonal, with compact support, and their connection coefficients can be computed. The method developed by Latto et al. to compute connection coefficients does not provide the correct inner product near the endpoints of a bounded interval, making the implementation of boundary conditions problematic. Moreover, the highly oscillatory nature of the wavelet basis functions makes standard numerical quadrature of integrals near the boundary impractical. The authors extend the method of Latto et al. to construct and solve a linear system of equations whose solution provides the exact computation of the integrals at the boundaries. As a consequence, they provide the correct inner product for wavelet basis functions on a bounded interval.