A General Hilbert Space Approach to Framelets

Abstract

In arbitrary separable Hilbert spaces it is possible to deffine multiscale methods of constructive approximation based on product kernels, restricting their choice in certain ways. These wavelet techniques have already filtering and localization properties and they are applicable in many areas due to their generalized deffinition. But they lack detailed information about their stability and redundancy, which are frame properties. So in this work frame conditions are introduced for approximation methods based on product kernels. In order to provide stability and redundancy the choice of product kernel ansatz function has to be restricted. Taking into account the kernel conditions for multiscale and for frame approximations one is able to deffine wavelet frames (= framelets), inheriting the approximation properties of both techniques and providing a more precise tool for multiscale analysis than the normal wavelets.