Page 130 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 130
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
as a graph analog of radial basis function methods in euclidean spaces or on the sphere. We
provide several descriptions of positive definite functions on graphs, the most relevant one is
a Bochner- type characterization in terms of positive Fourier coefficients. These descriptions
allow us to design GBF’s and to study GBF approximation in more detail: we are able to char-
acterize the native spaces of the interpolants, we give explicit estimates for the approximation
error and provide ways on how to calculate the approximants in an efficient manner. As a final
application, we show how GBFs can be used for classification tasks on graphs.
Strict positive definiteness of non-radial kernels on d-dimensional spheres
Janin Jäger, janin.jaeger@math.uni-giessen.de
Justus Liebig University, Germany
Coauthor: Martin Buhmann
Isotropic positive definite functions are used in approximation theory and are for example ap-
plied in geostatistics and physiology. They are also of importance in statistics where they occur
as correlation functions of homogeneous random fields on spheres. We study a class of func-
tion applicable for interpolation of arbitrary scattered data on Sd−1 by linear combinations of a
kernel K : Sd−1 × Sd−1 → C evaluated at the interpolation points in the second argument. The
isotropic kernels are a special case of this approach and we study kernels with more general
properties like axial symmetry and invariance under parity.
A class of kernels for which the resulting interpolation problem is uniquely solvable for
any distinct point set Ξ ⊂ Sd are known strict positive definite isotropic functions. Using recent
results of Bonfim and Menegatto [2] and the famous representations of isotropic positive definite
functions on Sd−1 due to Schoenberg as starting point we derive new sufficient conditions for
strict positive definiteness of axial symmetric and convolutional kernels. The results extend a
necessary and sufficient characterisation of strict positive definite isotropic basis functions by
Chen et al. proven in [1] to a non-radial kernel class.
References
[1] Chen, D. and Menegatto, V. A. and Sun, X.: A necessary and sufficient condition for
strictly positive definite functions on spheres, Proceedings of the American Mathematical
Society, 131, 2733-2740, 2003.
[2] Bonfim, R. and Menegatto, V., Strict positive definiteness of multivariate covariance
functions on compact two-point homogeneous spaces, Journal of Multivariate Analysis,
152, 2016.
Admissible meshes on algebraic sets
Agnieszka Kowalska, agnieszka.kowalska@up.krakow.pl
Pedagogical University of Krakow, Poland
Coauthor: Leokadia Białas-Ciez˙
Admissible meshes were formally introduced by J. P. Calvi and N. Levenberg in 2008. Such
meshes are nearly optimal for uniform least squares approximation and contain interpolation
128
as a graph analog of radial basis function methods in euclidean spaces or on the sphere. We
provide several descriptions of positive definite functions on graphs, the most relevant one is
a Bochner- type characterization in terms of positive Fourier coefficients. These descriptions
allow us to design GBF’s and to study GBF approximation in more detail: we are able to char-
acterize the native spaces of the interpolants, we give explicit estimates for the approximation
error and provide ways on how to calculate the approximants in an efficient manner. As a final
application, we show how GBFs can be used for classification tasks on graphs.
Strict positive definiteness of non-radial kernels on d-dimensional spheres
Janin Jäger, janin.jaeger@math.uni-giessen.de
Justus Liebig University, Germany
Coauthor: Martin Buhmann
Isotropic positive definite functions are used in approximation theory and are for example ap-
plied in geostatistics and physiology. They are also of importance in statistics where they occur
as correlation functions of homogeneous random fields on spheres. We study a class of func-
tion applicable for interpolation of arbitrary scattered data on Sd−1 by linear combinations of a
kernel K : Sd−1 × Sd−1 → C evaluated at the interpolation points in the second argument. The
isotropic kernels are a special case of this approach and we study kernels with more general
properties like axial symmetry and invariance under parity.
A class of kernels for which the resulting interpolation problem is uniquely solvable for
any distinct point set Ξ ⊂ Sd are known strict positive definite isotropic functions. Using recent
results of Bonfim and Menegatto [2] and the famous representations of isotropic positive definite
functions on Sd−1 due to Schoenberg as starting point we derive new sufficient conditions for
strict positive definiteness of axial symmetric and convolutional kernels. The results extend a
necessary and sufficient characterisation of strict positive definite isotropic basis functions by
Chen et al. proven in [1] to a non-radial kernel class.
References
[1] Chen, D. and Menegatto, V. A. and Sun, X.: A necessary and sufficient condition for
strictly positive definite functions on spheres, Proceedings of the American Mathematical
Society, 131, 2733-2740, 2003.
[2] Bonfim, R. and Menegatto, V., Strict positive definiteness of multivariate covariance
functions on compact two-point homogeneous spaces, Journal of Multivariate Analysis,
152, 2016.
Admissible meshes on algebraic sets
Agnieszka Kowalska, agnieszka.kowalska@up.krakow.pl
Pedagogical University of Krakow, Poland
Coauthor: Leokadia Białas-Ciez˙
Admissible meshes were formally introduced by J. P. Calvi and N. Levenberg in 2008. Such
meshes are nearly optimal for uniform least squares approximation and contain interpolation
128
