NDLI: Approximation of eigenfunctions in kernel-based spaces

Content Provider	Springer Nature Link
Author	Santin, G. Schaback, R.
Copyright Year	2016
Abstract	Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the “native” Hilbert space $\mathcal {H}$ in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L 2-orthonormal and orthogonal in $\mathcal {H}$ (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of $\mathcal {H}$ . These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.
Ending Page	993
Page Count	21
Starting Page	973
File Format	PDF
ISSN	10197168
e-ISSN	15729044
Journal	Advances in Computational Mathematics
Issue Number	4
Volume Number	42
Language	English
Publisher	Springer US
Publisher Date	2016-01-21
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Visualization Radial basis functions Computational Mathematics and Numerical Analysis Mathematical and Computational Biology Approximations and expansions Eigenfunctions Integral transforms Approximation by arbitrary nonlinear expressions; widths and entropy Computational Science and Engineering n-widths Optimal subspaces (Generalized) eigenfunction expansions; rigged Hilbert spaces Interpolation Integral operators Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) Mercer kernels Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) Eigenvalues Nontrigonometric harmonic analysis Eigenvalues, eigenvectors General harmonic expansions, frames Mathematical Modeling and Industrial Mathematics Greedy methods Positive definite functions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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