Translates of exponential box splines and their related spaces
Translates of exponential box splines and their related spaces
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指数箱样条及其相关空间的平移
DOI：
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
A. Ron
中科院分区：
文献类型：
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作者：
A. Ben;A. Ron
指数箱样条（EB样条）是规则网格上的多元紧支撑函数，它在指数多项式张成的空间Z中分段。这个空间可以定义为某些常系数偏微分算子的核的交集。本文主要讨论EB样条的整数平移所张成的整函数空间H的代数分析。这项研究依赖于Z及其离散模拟S°的详细描述。这里所采取的方法是基于这样的观察，即当Z由纯指数跨越时，Z的结构相对简单，而所有其他情况都可以借助于适当的限制过程来分析。此外，我们发现它更有效地直接应用相关的微分和差分算子，而不是傅立叶分析的替代技术。因此，在推广已知的多项式箱样条理论的同时，这里的结果提供了一种更简单的方法和对这一重要特殊情况的新见解。我们还确定和详细研究了几种类型的奇点只发生在复杂的EB样条。第一种情况是当EB样条的傅里叶变换在某些临界点处为零时，第二种情况是当Z不能嵌入y时，第三种情况是当H是Z的真子空间时。我们表明，除其他外，这三种情况下的每一个严格包括在其前，他们都可以避免细化的网格。
Exponential box splines (EB-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space Z spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space H of all entire functions spanned by the integer translates of an EB-spline. This investigation relies on a detailed description of Z and its discrete analog S°. The approach taken here is based on the observation that the structure of Z is relatively simple when Z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex EB-splines. The first is when the Fourier transform of the EB-spline vanishes at some critical points, the second is when Z cannot be embedded in y and the third is when H is a proper subspace of Z. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.