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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
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