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Many types of radial basis functions (RBFs) are global, in terms of having large magnitude across the entire domain. Yet, in contrast, for example, with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of... more
Many types of radial basis functions (RBFs) are global, in terms of having large magnitude across the entire domain. Yet, in contrast, for example, with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coecients. That is, changing a single data value mainly aects the coecients of the RBFs which are centered in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coecients for cardinal data, in both 1-D and 2-D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs. Key words. Radial basis functions, RBF, cardinal interpolation AMS subject classifications. 41A05, 41A58, 41A60, 41A63, 65D05 1. Introduction. Radial basis functions (RBF) provide a well established ap- proach to the task of interpolating scattered data in multiple dimensions. With a radial function ( r ) and with data values f k given at locations x k , k = 1 , 2 ,...,n , the function s ( x ) = n X k =1 k ( || x x k ||
... Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering (iMMC), Place du Levant 1, 1348 Louvain-la-Neuve, Belgium cecile.piret@uclouvain.be, jean-francois.remacle@uclouvain.be,... more
... Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering (iMMC), Place du Levant 1, 1348 Louvain-la-Neuve, Belgium cecile.piret@uclouvain.be, jean-francois.remacle@uclouvain.be, emilie.marchandise@uclouvain.be Summary. ...
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