Gaussian radial‐basis functions: Cardinal interpolation of $\backslash ell^p$ and power‐growth data

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• 作者：Riemenschneider ; S.D. ; Sivakumar ; N.
• 刊名：Advances in Computational Mathematics
• 出版时间：1999
• 出版年：1999
• 期刊代码：1_10197168
• 类别：mt
• 卷：11
• 期：2/3
• 页码：229-251
• 数据来源：sp

摘要

Suppose $\backslash lamb$da is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function $L\_\backslash lamb$da (x)=\sum_{k\in\mathbb{Z}} c_k \exp({-}\lambda(x-k)^2), $x\backslash in\backslash mathbb\{R\}$, satisfying the interpolatory conditions $L\_\backslash lamb$da (j)=\delta_{0j}, $j\backslash in\backslash mathbb\{Z\}$. The paper considers the Gaussian cardinal interpolation operator $(\backslash mathcal\{L\}\_\backslash lamb$da \mathbf{y})(x):=\sum_{k\in\mathbb{Z}} y_k L_\lambda (x-k), \quad \mathbf{y}=(y_k)_{k\in\mathbb{Z}}, \ x\in\mathbb{R}, as a linear mapping from $\backslash ell^p(\backslash mathbb\{Z\})$ into $L^p(\backslash mathbb\{R\})$, , and in particular, its behaviour as $\backslash lamb$da\to0^+. It is shown that $\backslash Vert\; \backslash mathcal\{L\}\_\backslash lamb$da \Vert _p is uniformly bounded (in $\backslash lamb$da) for d that $\backslash Vert\; \backslash mathcal\{L\}\_\backslash lamb$da \Vert _1\asymp \log(1/\lambda) as $\backslash lamb$da\to0^+. The limiting behaviour is seen to be that of the classical Whittaker operator $\backslash mathcal\{W\}\backslash$dvtx \mathbf{y}\mapsto \sum_{k\in\mathbb{Z}} y_k\,{\sin\pi(x-k) \over \pi(x-k)}, in that $\backslash lim\_\{\backslash lamb$da\to0^+}\Vert \mathcal{L}_\lambda \mathbf{y}-\mathcal{W}\mathbf{y}\Vert _p=0, for every $\backslash mathbf\{y\}\backslash in\backslash ell^p(\backslash mathbb\{Z\})$ and dinal interpolants to a function $f$ which is the Fourier transform of a tempered distribution supported in $(\{-\}\backslash pi,\backslash pi)$ converge locally uniformly to $f$ as $\backslash lamb$da\to0^+. Multidimensional extensions of these results are also discussed.