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Recent Results on Rational Approximation and Interpolation with Completely and Multiply Monotone Radial Basis Functions

Presented by: 
Martin Buhmann
Wednesday 20th February 2019 - 11:00 to 11:35
INI Seminar Room 1
We will report on new results about approximations to continuous functions of multiple variables. We shall use either approximation with interpolation or approximation by rational functions. For these kinds of approximations, radial basis functions are particularly attractive, as they provide regular, positive definite or conditionally positive definite approximations, independent of the spatial dimension and independent the distribution of the data points we wish to work with. These interpolants have very many applications for example in solving nonlinear partial differential equations by collocation. In this talk, we classify radial basis and other functions that are useful for such scattered data interpolation or for rational approximations from vector spaces spanned by translates of those basis functions (kernels); for this we study in particular multiply and/or completely monotone functions. We collect special properties of such monotone functions, generalise them and find larger classes than the well known monotone functions for multivariate interpolation. Furthermore, we discuss efficient ways to compute rational approximations using the same type of kernels.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons