1.    Approximation of Functions & Wavelets

 A common approach for numerical methods is to describe the numerical solution as a linear combination of basis functions.  These should allow for accurate approximations to the true solution by means of a small number of  actually used basis functions . Wavelets (or  multiscale basisfunctions)  do!  Two particular univariate examples of such multiscale bases are  the Haar-wavelets (left) and the Hierarchical basis (right) below:    Fig. 1    wavelets  of  the first three levels for the so-called Haar- and the Hierarchical basis Multiscale functions are denoted by  .    Here  l  is the  level index and   t  the index for the spatial translation. The number of  mutiscale functions on level l  is typically  something like O(2^l ). The main reason for the approximation power of multiscale bases is that just a few basis functions from low (high) levels with large (small)  support are sufficient  to approximate  the rather global smooth  (localized non-smooth)  parts of the target function.  In the below example a function with steep gradients is approximated by a superposition of just four hat functions from the first four refinement levels.   The approximation power of the Haar- and Hierarchical Basis is rather weak. More favourable are high order generalizations of  the hat functions: Interpolets [5]. They are smoother and they reproduce polynomials of higher degree.