Multi- and Infinite-variate Integration and $L^2$-Approximation on Hilbert Spaces with Gaussian Kernels
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915654874955776 |
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| author | Gnewuch, Michael Ritter, Klaus Rüßmann, Robin |
| author_facet | Gnewuch, Michael Ritter, Klaus Rüßmann, Robin |
| contents | We study integration and $L^2$-approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor product form. In the infinite-variate case, for both computational problems, we establish matching upper and lower bounds for the polynomial convergence rate of the $n$-th minimal error. In the multivariate case, we improve several tractability results for the integration problem. For the proofs, we establish the following transference result together with an explicit construction: Each of the computational problems on a space with a Gaussian kernel is equivalent on the level of algorithms to the same problem on a Hermite space with suitable parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi- and Infinite-variate Integration and $L^2$-Approximation on Hilbert Spaces with Gaussian Kernels Gnewuch, Michael Ritter, Klaus Rüßmann, Robin Numerical Analysis 65D30, 65D32, 65Y20, 68Q17 We study integration and $L^2$-approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor product form. In the infinite-variate case, for both computational problems, we establish matching upper and lower bounds for the polynomial convergence rate of the $n$-th minimal error. In the multivariate case, we improve several tractability results for the integration problem. For the proofs, we establish the following transference result together with an explicit construction: Each of the computational problems on a space with a Gaussian kernel is equivalent on the level of algorithms to the same problem on a Hermite space with suitable parameters. |
| title | Multi- and Infinite-variate Integration and $L^2$-Approximation on Hilbert Spaces with Gaussian Kernels |
| topic | Numerical Analysis 65D30, 65D32, 65Y20, 68Q17 |
| url | https://arxiv.org/abs/2412.05368 |