Orthonormal wavelet bases adapted for partial differential equations with boundary conditions

We adapt ideas presented by Auscher to impose boundary conditions on the construction of multiresolution analyses on the interval, as introduced by Cohen, Daubechies, and Vial. We construct new orthonormal wavelet bases on the interval satisfying homogeneous boundary conditions. This construction can be extended to wavelet packets in the case of one boundary condition at each edge. We present in detail the numerical computation of the filters and the derivative operators associated with these bases. We derive quadrature formulae in order to study the approximation error at the edge of the interval. Several examples illustrate the present construction.