Approximate Wavelets and the Approximation of Pseudodifferential Operators

This paper studiesapproximate multiresolution analysisfor spaces generated by smooth functions providing high-order semi-analytic cubature formulas for multidimensional integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy, methods from wavelet theory can be applied. We obtain an approximate decomposition of the finest scale space into almost orthogonal wavelet spaces. For the example of the Gaussian function we study some properties of the analytic prewavelets and describe the projection operators onto the wavelet spaces. The multivariate wavelets retain the property of the scaling function to provide efficient analytic expressions for the action of important integral operators, which leads to sparse and semi-analytic representations of these operators.