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Radial Basis Functions

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Encyclopedia of Applied and Computational Mathematics

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Mathematics Subject Classification

41A05; 41A15; 41A30; 65D05; 65D07; 65D15

Short Definition of Radial Basis Functions

Approximations using radial basis functions are multivariate kernel methods to approximate multivariable functions by finite linear combinations of translates of a single, univariate, quasi-stationary function (the “radial basis function”). Before translated, it is composed with the Euclidean norm so that it is rotationally invariant and may thus be used in any dimension. They are usually means to approximate functions which are only known at a finite number of points (“centres”), in order that numerous evaluations of the approximating function at other points can be made efficiently later on.

Applications include computer-aided geometric design, neural networks, and supervised or unsupervised learning, for instance by support vector machines [5]. The data dependence of translates opens the door to existence and uniqueness theorems for interpolating problems at...

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Justus-Liebig-Universität, Giessen, Germany

    Martin Buhmann

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  1. Martin Buhmann
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Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Buhmann, M. (2015). Radial Basis Functions. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_306

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