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Regularized Kernel-Based Reconstruction in Generalized {B}esov Spaces

: Griebel, M.; Rieger, C.; Zwicknagl, B.

Fulltext ()

Foundations of Computational Mathematics 18 (2018), No.2, pp.459-508
ISSN: 1615-3375
ISSN: 1615-3383
Journal Article, Electronic Publication
Fraunhofer SCAI ()

We present a theoretical framework for reproducing kernel-based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set.