Countable tensor products of Hermite spaces and spaces of Gaussian kernels ?
Autor(en): | Gnewuch, M. Hefter, M. Hinrichs, A. Ritter, K. |
Stichwörter: | ALGORITHMS; ANOVA decompostion; APPROXIMATION; Computer Science; Computer Science, Theory & Methods; Countable tensor products; EFFICIENT; Gaussian kernels; Hermite polynomials; Hermite spaces; HILBERT-SPACES; INFINITE-DIMENSIONAL INTEGRATION; Mathematics; Mathematics, Applied; Reproducing kernel Hilbert spaces | Erscheinungsdatum: | 2022 | Herausgeber: | ACADEMIC PRESS INC ELSEVIER SCIENCE | Journal: | JOURNAL OF COMPLEXITY | Volumen: | 71 | Zusammenfassung: | In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete tensor product in the sense of von Neumann may be identified with an RKHS whose domain is a proper subset of the sequence space RN. Moreover, we show that each tensor product of spaces of Gaussian kernels having squaresummable shape parameters is isometrically isomorphic to a tensor product of Hermite spaces; the corresponding isomorphism is given explicitly, respects point evaluations, and is also an L2isometry. This result directly transfers to the case of finite tensor products. Furthermore, we provide regularity results for Hermite spaces of functions of a single variable.(c) 2022 Elsevier Inc. All rights reserved. |
ISSN: | 0885-064X | DOI: | 10.1016/j.jco.2022.101654 |
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geprüft am 14.05.2024