Next: Trial Spaces & Adaptivity Up: Mathematical Introduction Previous: Univariate Wavelets

# Multivariate Wavelets

AWFD supports tensor product type multivariate wavelets only. Therefore, the underlying computational domain is always a multi-dimensional rectangular box, by default. There are two types of tensor product wavelets. The first construction defines the wavelets by

Here and in the remainder of this document we use bold face letters to denote multi-indices. (The index range starts with 0 to be consistent with C++ convention.) The above approach is algorithmically very simple and leads to basis functions with remarkable approximation properties (->sparse grid effect). The wavelets may have strongly anisotropic supports, depending on how much the level indices differ. This explains, why we call them anisotropic tensor product wavelets.

The second (and most common) approach to multivariate wavelets is the Meyer-construction, which was meant to preserve a linear order of multi-resolution spaces and (almost) isotropic supports of the wavelets. The wavelets are defined by

For the isotropic wavelets there are only a few functions implemented in AWFD, namely a wavelet transform and an inverse wavelet transform for uniform grids. In section 9.1 there is an example how to call these functions.

Next: Trial Spaces & Adaptivity Up: Mathematical Introduction Previous: Univariate Wavelets
koster 2003-07-29