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The asymptotic cardinal function of the multiquadratic $$\varphi{}(r)=(r^ 2+c^ 2)^{1/2}$$ as $$c{\rightarrow{}}\infty$$. (English) Zbl 0764.41016
Summary: A radial basis function approximation has the form $s(x)=\sum_{j\in\mathbb{Z}^ d} y_ j\varphi(\| x-x_ j\|_ 2), \qquad x\in\mathbb{R}^ d,$ where $$\varphi: [0,\infty)\to\mathbb{R}$$ is some given function, $$(y_ j)_{j\in\mathbb{Z}^ d}$$ are real coefficients, and the centres $$(x_ j)_{j\in\mathbb{Z}^ d}$$ are points in $$\mathbb{R}^ d$$. It is known that radial basis function approximations using the multiquadratic $$\varphi(r)=(r^ 2+c^ 2)^{1/2}$$ possess many useful and interesting properties when the centres form an infinite regular lattice. We analyse the limiting case as $$c\to\infty$$ and identify a class of functions that arise as uniform limits of the multiquadric interpolants. In the univariate case, we observe that the cardinal function for the multiquadric becomes the sinc function as $$c\to\infty$$. The limit of the multivariate cardinal function is also identified.

##### MSC:
 41A30 Approximation by other special function classes
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##### References:
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