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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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