Gaussian Quadratures with prescribed nodes via moment theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929651697319936 |
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| author | Nailwal, Rajkamal Zalar, Aljaž |
| author_facet | Nailwal, Rajkamal Zalar, Aljaž |
| contents | Let $μ$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $μ$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $μ$ in terms of the roots of a bivariate polynomial is given and two determinantal representations of this polynomial are established. In particular, the authors solved the question of the existence of a minimal quadrature rule with one prescribed node, leaving open the extension to more prescribed nodes. In this paper, we solve this problem using moment theory as the main tool. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_20849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian Quadratures with prescribed nodes via moment theory Nailwal, Rajkamal Zalar, Aljaž Functional Analysis Numerical Analysis 65D32, 47A57, 47A20, 44A60 (Primary), 15A04, 47N40 (Secondary) Let $μ$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $μ$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $μ$ in terms of the roots of a bivariate polynomial is given and two determinantal representations of this polynomial are established. In particular, the authors solved the question of the existence of a minimal quadrature rule with one prescribed node, leaving open the extension to more prescribed nodes. In this paper, we solve this problem using moment theory as the main tool. |
| title | Gaussian Quadratures with prescribed nodes via moment theory |
| topic | Functional Analysis Numerical Analysis 65D32, 47A57, 47A20, 44A60 (Primary), 15A04, 47N40 (Secondary) |
| url | https://arxiv.org/abs/2412.20849 |