Convergence properties of radial basis functions
Convergence properties of radial basis functions
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DOI：
10.1007/bf02075461
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发表时间：
1988-12
影响因子：
2.7
通讯作者：
I. R. H. Jackson
中科院分区：
文献类型：
--
作者：
I. R. H. Jackson
多元插值问题是在不同的数据点xi上选择一个满足插值条件的函数。采用径向基函数φ:R+→R，意味着它来自函数所张成的线性空间，除了有时会添加低阶多项式和施加额外自由度的进一步条件。michelli[6]表明，对于某些选择，这些条件唯一地定义了。然而，现在要考虑的是更基本的空间适宜性问题。给出了线性空间的成员在有界域上局部一致收敛于规定的连续函数的充分条件，如数据点{xi:i=1,2，…变得稠密。对于ϕ(r)=r的情况，发现当n为奇数时它们成立，而当n为偶数时则不成立。
The multivariate interpolation problem is that of choosing a functionsfromR“ toRthat satisfies the interpolation conditionsatmdifferent data points xi. Employing the radial basis functionϕ:R+→R, means thatsis from the linear space spanned by the functionsexcept that low-order polynomials are sometimes added and further conditions imposed onsthat take up the extra degrees of freedom. Micchelli [6] shows that for some choices ofϕthese conditions uniquely defines. Now, however, the more basic question of the suitability of the spaces is considered. Sufficient conditions are given for members of the linear space to converge locally uniformly to a prescribed continuous function on a bounded domain, as the data points {xi:i=1,2,...} become dense. For the caseϕ(r)=rthey are found to hold when n is odd but not whennis even.