On the Approximation of Singular Functions by Series of Non-integer Powers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913601472692224 |
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| author | Zhao, Mohan Serkh, Kirill |
| author_facet | Zhao, Mohan Serkh, Kirill |
| contents | In this paper, we describe an algorithm for approximating functions of the form $f(x)=\int_{a}^{b} x^μ σ(μ) \, d μ$ over $[0,1]$, where $σ(μ)$ is some signed Radon measure, or, more generally, of the form $f(x) = <σ(μ),\, x^μ>$, where $σ(μ)$ is some distribution supported on $[a,b]$, with $0 <a < b < \infty$. One example from this class of functions is $x^c (\log{x})^m=(-1)^m <δ^{(m)}(μ-c), \, x^μ>$, where $a\leq c \leq b$ and $m \geq 0$ is an integer. Given the desired accuracy $ε$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $t_1$, $t_2$, $\ldots$, $t_N$, so that the functions are approximated by series of the form $f(x)\approx \sum_{j=1}^N c_j x^{t_j}$, and a set of collocation points $x_1$, $x_2$, $\ldots$, $x_N$, such that the expansion coefficients can be found by collocating the function at these points. We prove that our method has a small uniform approximation error which is proportional to $ε$ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}ε})$. We demonstrate the performance of our algorithm with several numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_10439 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Approximation of Singular Functions by Series of Non-integer Powers Zhao, Mohan Serkh, Kirill Numerical Analysis 41A30, 41A58, 65D15 In this paper, we describe an algorithm for approximating functions of the form $f(x)=\int_{a}^{b} x^μ σ(μ) \, d μ$ over $[0,1]$, where $σ(μ)$ is some signed Radon measure, or, more generally, of the form $f(x) = <σ(μ),\, x^μ>$, where $σ(μ)$ is some distribution supported on $[a,b]$, with $0 <a < b < \infty$. One example from this class of functions is $x^c (\log{x})^m=(-1)^m <δ^{(m)}(μ-c), \, x^μ>$, where $a\leq c \leq b$ and $m \geq 0$ is an integer. Given the desired accuracy $ε$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $t_1$, $t_2$, $\ldots$, $t_N$, so that the functions are approximated by series of the form $f(x)\approx \sum_{j=1}^N c_j x^{t_j}$, and a set of collocation points $x_1$, $x_2$, $\ldots$, $x_N$, such that the expansion coefficients can be found by collocating the function at these points. We prove that our method has a small uniform approximation error which is proportional to $ε$ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}ε})$. We demonstrate the performance of our algorithm with several numerical experiments. |
| title | On the Approximation of Singular Functions by Series of Non-integer Powers |
| topic | Numerical Analysis 41A30, 41A58, 65D15 |
| url | https://arxiv.org/abs/2308.10439 |