Error bounds for function approximation using generated sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909598244405248 |
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| author | Cools, Ronald Nuyens, Dirk Wilkes, Laurence |
| author_facet | Cools, Ronald Nuyens, Dirk Wilkes, Laurence |
| contents | This paper explores the use of "generated sets" $\{ \{ k \boldsymbolζ \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist $\boldsymbolζ \in [0,1]^d$ for which the worst-case $L_2$ error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error bounds for function approximation using generated sets Cools, Ronald Nuyens, Dirk Wilkes, Laurence Numerical Analysis 65D15, 65T40 This paper explores the use of "generated sets" $\{ \{ k \boldsymbolζ \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist $\boldsymbolζ \in [0,1]^d$ for which the worst-case $L_2$ error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results. |
| title | Error bounds for function approximation using generated sets |
| topic | Numerical Analysis 65D15, 65T40 |
| url | https://arxiv.org/abs/2505.00440 |