On the convergence of cardinal interpolation by parameterized radial basis functions

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• 作者：Martin D. Buhmann^a ; ^{martin.buhmann@math.uni-giessen.de} ; Han Feng^b ; ^c ; ^{hfeng3@ualberta.ca}
• 关键词：Cardinal interpolation ; Radial basis function ; Band limited ; Multiquadric
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 May 2017
• 年：2017
• 卷：449
• 期：1
• 页码：718-733
• 全文大小：389 K
• 卷排序：449

文摘

We consider cardinal interpolation on gridded data by using various radial basis functions associated with one or two parameters, one of which leads asymptotically to so-called flat-limits. Previously it had been shown that the classical Paley–Wiener functions can be recovered by such cardinal interpolations as the parameter tends to infinity. In this article, we extend the results by relaxing the requirements on the approximand functions from several points of view. The radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics, inverse multiquadrics and shifted thin-plate spline radial basis functions for instance. We also generalise the classes of admitted approximands as well as the radial basis functions to generalised multiquadrics in place of the well-known ordinary or for example inverse multiquadrics. An interesting analytical aspect of this work is that – unlike the classical Whittaker–Shannon theorem – functions (approximands) may be reproduced for the parameter c→∞c→∞ in the generalised multiquadrics cardinal approximands, where the usual Shannon series does not converge with theses approximands due to the slow decay of the sinc-function which does not allow e.g. polynomials as approximands. In contrast to the latter, the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter c that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity.