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On the local linear independence of translates of a box spline. (English) Zbl 0545.41018
For any $$s\times n$$ matrix X of rank s the box spline $$B(x| X)$$ is a piecewise polynomial of degree n-s defined by requiring that $$\int_{{\mathbb{R}}^ s}f(x)B(x| X)dx=\int_{[0,1]^ n}f(Xu)du$$ holds for any continuous function f on $${\mathbb{R}}^ s$$. Assuming $$X\subset Z^ s$$ (where the set of columns of X is also denoted by X) the central objective of this paper is to determine under which circumstances the translates $$B(\cdot -\alpha | X)$$ are locally linearly independent. The approach to this problem relies on studying the space of distributions $$D(X)=\{f:\quad D_ Vf=0,\quad \forall V\subset X\ni<X\backslash V>\neq {\mathbb{R}}^ s\}$$ where $$<V>=span\{V\}$$, $$D_ vf=\sum^{s}_{j=1}v_ j\partial f/\partial x_ j,\quad D_ Vf=(\prod_{v\in V}D_ v)f.$$ D(X) consists of polynomials of degree at most n-s on $${\mathbb{R}}^ s$$ and the first main result is that $$\dim D(X)=\#{\mathcal B}(X)$$ where $${\mathcal B}(X)=\{Y\subset X:$$ #Y$$=s$$, $$<Y>={\mathbb{R}}^ s\}$$. In view of the fact that D(X) is spanned by the polynomial pieces of $$B(\cdot | X)$$ the question of local linear independence of the $$B(\cdot -\alpha | X)$$ reduces to comparing dim D(X) with #$$b(x| X)$$ where for any generic point $$x\in {\mathbb{R}}^ s b(x| X)=\{\alpha \in Z^ s:$$ B(x-$$\alpha | X)\neq 0\}$$. To this end, it is shown that $$\#b(x| X)=\sum_{Y\in {\mathcal B}(X)}| \det Y|.$$ Thus the $$B(\cdot -\alpha | X)$$, $$\alpha \in Z^ s$$, are locally linearly independent if and only if $$| \det Y| =1$$, $$Y\in {\mathcal B}(X)$$, i.e., (X,$${\mathcal B}(X))$$ can be viewed as a regular represented matroid. Furthermore, it is shown that in this case the set $$b(x| X)$$ is unisolvent for interpolation by polynomials in D(X). This in turn is used to construct for any p, 1$$\leq p\leq \infty$$, linear projectors from $$L_ p(\Omega)$$, $$\Omega$$ any domain in $${\mathbb{R}}^ s$$, onto span $$\{B(\cdot -\alpha | X)_{| \Omega}: \alpha \in Z^ s\}$$.

##### MSC:
 41A15 Spline approximation 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 05B35 Combinatorial aspects of matroids and geometric lattices 05B40 Combinatorial aspects of packing and covering
##### Keywords:
box spline; piecewise polynomial; space of distributions
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