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 extanded to wavelet packets, in the case of one boundary condition at each edge. We present in detail the numerical computation of the filters and of the derivative operators associated to 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. Key words : wavelet, multiresolution analysis, boundary conditions. AMS subject classification : 35C10, 42C15, 46E35. WAVELETS WITH BOUNDARY CONDITIONS 2 Contents 1 Introduction 3 2 Orthonormal wavelet bases on IR 4 3 Orthonormal wavelet bases on [0; +1[ 5 3.1 Multiresolution analysis on [0; +1[ without boundary conditions : : : : ..