On Shifted Cardinal Interpolation by Gaussians and Multiquadrics
On Shifted Cardinal Interpolation by Gaussians and Multiquadrics
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关于高斯和多重二次曲面的平移基数插值
DOI：
10.1006/jath.1996.0091
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发表时间：
1996
影响因子：
0.9
通讯作者：
N. Sivakumar
中科院分区：
文献类型：
--
作者：
B. Baxter;N. Sivakumar
径向基函数近似是固定函数R的平移的线性组合：R d R。这样的函数具有许多有用的和有趣的性质时，translates是整数和径向对称。我们研究了与此密切相关的问题，其中固定函数是移位高斯函数G =G（x），其中G（x）=exp（x × 22）和Rd。具体地说，我们利用椭圆函数的理论来建立Toeplitz算子公式]的可逆性，当它没有半整数分量时;否则它是奇异的。这意味着存在移位高斯基数函数，即移位高斯的整数平移的线性组合满足k（j）= k 0 j。我们还研究了当参数趋于零时的移位基数函数。特别地，当移位向量不含半整数分量时，我们发现它们一致收敛于sinc函数。我们的方法是基于部分类似的结果建立的第一作者时，基函数是哈代multiquadric。在最后一章中描述了与移位B样条基数插值理论的几个有趣的联系。
A radial basis function approximation is a linear combination of translates of a fixed function�:�Rd�R. Such functions possess many useful and interesting properties when the translates are integers and�is radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussian�=G(���), whereG(x)=exp(���x�22) and��Rd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operatorformula]when�has no half-integer components; it is singular otherwise. This implies the existence of ashifted Gaussian cardinal function, that is, a linear combination�of integer translates of the shifted Gaussian satisfying�(j)=�0j. We also study shifted cardinal functions when the parameter�tends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vector�possesses no half-integer components. Our methods are based in part on similar results established by the first author when the basis function is the Hardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale.