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Multivariate Wavelets

AWFD supports tensor product type multivariate wavelets only. Therefore, the underlying computational domain is always a multi-dimensional rectangular box, $[0,1]^D$ by default. There are two types of tensor product wavelets. The first construction defines the wavelets by

$$\psi_{\bf l,t}({\bf x}) := \prod_{i=0}^{D-1} \psi_{l_i,t_i}(... ...}), {\bf l}=(l_0,...,l_{D-1}) \quad\quad\mbox{(anisotropic)} .$$

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 $l_0,...,l_{D-1}$ 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

$$\psi_{l,{\bf t,e}}({\bf x}):=\prod_{i=0}^{D-1} \psi^{e_i}_{l... ...nd }\quad \psi^1_{l,t}:=\psi_{l,t}\quad\quad\mbox{(isotropic)}$$

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.