Compactly supported tight affine spline frames in ${L}_{2}\left({\mathbb{R}}^{d}\right)$.

*(English)*Zbl 0892.42018Summary: The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in ${L}_{2}\left({\mathbb{R}}^{d}\right)$ 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 $\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$.

##### MSC:

42C15 | General harmonic expansions, frames |

41A15 | Spline approximation |

42C30 | Completeness of sets of functions of non-trigonometric Fourier analysis |

41A63 | Multidimensional approximation problems |