Multiresolution Inverse Wavelet Reconstruction from a Fourier Partial Sum

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the dis-continuities. In previous work [11,12] we described a numerical procedure for overcoming the Gibbs phenomenon called the Inverse Wavelet Reconstruction method (IWR). The method takes the Fourier coe¢ cients of an oscillatory partial sum and uses them to construct the wavelet coe¢ cients of a non-oscillatory wavelet series. However, we only described the method standard wavelet series and not for the continuous wavelet transform. We also only described the method for so-called crude wavelets: wavelets which do not admit a scaling function, and therefore lack a multiresolution analysis. Here we describe a version of IWR based on a quadrature discretization of the continuous inverse wavelet transform. We also describe the multiresolution potential of the Inverse wavelet reconstruction method, which is the ability to decompose a function into a coarse trend plus …ner details.