Compactly supported tight affine spline frames in $L_{2}(\mathbb{R}^{d})$

The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in $L_{2}(\mathbb{R}^{d})$ from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of ``mother wavelets'', however, increases with the increase of the required smoothness.

Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix $\begin{pmatrix}1&1 1&-1\end{pmatrix}$.