Local linear independence of the translates of a box spline
Local linear independence of the translates of a box spline
复制标题
箱形样条平移的局部线性独立性
DOI：
10.1007/bf01890029
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发表时间：
1985
影响因子：
2.7
通讯作者：
R. Jia
中科院分区：
文献类型：
--
作者：
R. Jia
设n =（n）ln是Rm中的向量序列。箱样条函数定义为由下式给出的分布
$$M_\Xi：\varphi \to \int_{[0，1]^n } \varphi \left（{\sum\limits_{i = 1}^n {\lambda（i）\xi _i } } \right）d\lambda，\varphi \in C_c^\infty（R^m）.$$
.假设R包含Rm的基。则M ∈L∞（Rm）.假设
$$\Xi \子集V：= z^m .$$
.考虑平移Mv：=M E（·-v），v∈V。众所周知，（Mv）V是线性相关的，除非（*）
$$|\det Z| = 1forallbasesZ \subset \Xi$$
.本文证明了在条件（*）下，（Mv）V是局部线性无关的，即，
$$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$
在任何开集A上线性无关。
AbstractLet Ξ=(ξi)ln be a sequence of vectors inRm. The box splineMΞ is defined as the distribution given by
$$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$
. Suppose that Ξ contains a basis forRm. ThenMΞ∈L∞(Rm). Assume
$$\Xi \subset V: = z^m .$$
. Consider the translatesMv:=MΞ(·−v),v∈V. It is known that (Mv)V is linearly dependent unless(*)
$$|\det Z| = 1forallbasesZ \subset \Xi$$
. This paper demonstrates that under condition (*), (Mv)V is locally linearly independent, i.e.,
$$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$
is linearly independent over any open setA.