Box-Spline Tilings

Denote by $ B_n$ the $ n$-fold convolution of a $ d$-variate box-spline $ B$. Then, a function can be approximated in $ L_2(\mathbb{R}^d)$ by a sequence of cardinal splines

$\displaystyle p_n = \sum_{k\in\mathbb{Z}^d} c_{n,k} B_n(\cdot-k),\quad n=1,2,\ldots \,, $

if and only if the support of its Fourier transform is contained in the closure of the fundamental domain

$\displaystyle D: \left\vert\widehat{B}(x+2\pi k)\right\vert < \left\vert\widehat{B}(x)\right\vert,\quad \forall k\in\mathbb{Z}^d\backslash0 \,. $

The figure shows part of the tiling generated by $ D$ for the characteristic function $ B$ of the parallelogram spanned by the vectors $ (3,1)$ and $ (1,2)$.


$ \square$ with C. de Boor, Amer. Math. Monthly 98 (1991), 703-802.

$ \square$ with C. de Boor and S. Riemenschneider, Box-Splines, Springer, 1993.

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