{"title":"关于用非整数幂级数逼近奇异函数","authors":"Mohan Zhao, Kirill Serkh","doi":"10.1093/imanum/draf006","DOIUrl":null,"url":null,"abstract":"In this paper, we describe an algorithm for approximating functions of the form $f(x)=\\int _{a}^{b} x^{\\mu } \\sigma (\\mu ) \\, {\\text{d}} \\mu $ over $[0,1]$, where $\\sigma (\\mu )$ is some signed Radon measure, or, more generally, of the form $f(x) = {{\\langle \\sigma (\\mu ), x^\\mu \\rangle }}$, where $\\sigma (\\mu )$ is some distribution supported on $[a,b]$, with $0 <a < b< \\infty $. One example from this class of functions is $x^{c} (\\log{x})^{m}=(-1)^{m} {{\\langle \\delta ^{(m)}(\\mu -c), x^\\mu \\rangle }}$, where $a\\leq c \\leq b$ and $m \\geq 0$ is an integer. Given the desired accuracy $\\varepsilon $ and the values of $a$ and $b$, our method determines a priori a collection of noninteger powers $t_{1}$, $t_{2}$, …, $t_{N}$, 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 $x_{1}$, $x_{2}$, …, $x_{N}$, 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 $\\varepsilon $ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\\log{\\frac{1}{\\varepsilon }})$. We demonstrate the performance of our algorithm with several numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the approximation of singular functions by series of noninteger powers\",\"authors\":\"Mohan Zhao, Kirill Serkh\",\"doi\":\"10.1093/imanum/draf006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we describe an algorithm for approximating functions of the form $f(x)=\\\\int _{a}^{b} x^{\\\\mu } \\\\sigma (\\\\mu ) \\\\, {\\\\text{d}} \\\\mu $ over $[0,1]$, where $\\\\sigma (\\\\mu )$ is some signed Radon measure, or, more generally, of the form $f(x) = {{\\\\langle \\\\sigma (\\\\mu ), x^\\\\mu \\\\rangle }}$, where $\\\\sigma (\\\\mu )$ is some distribution supported on $[a,b]$, with $0 <a < b< \\\\infty $. One example from this class of functions is $x^{c} (\\\\log{x})^{m}=(-1)^{m} {{\\\\langle \\\\delta ^{(m)}(\\\\mu -c), x^\\\\mu \\\\rangle }}$, where $a\\\\leq c \\\\leq b$ and $m \\\\geq 0$ is an integer. Given the desired accuracy $\\\\varepsilon $ and the values of $a$ and $b$, our method determines a priori a collection of noninteger powers $t_{1}$, $t_{2}$, …, $t_{N}$, 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 $x_{1}$, $x_{2}$, …, $x_{N}$, 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 $\\\\varepsilon $ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\\\\log{\\\\frac{1}{\\\\varepsilon }})$. We demonstrate the performance of our algorithm with several numerical experiments.\",\"PeriodicalId\":56295,\"journal\":{\"name\":\"IMA Journal of Numerical Analysis\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IMA Journal of Numerical Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1093/imanum/draf006\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IMA Journal of Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imanum/draf006","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On the approximation of singular functions by series of noninteger powers
In this paper, we describe an algorithm for approximating functions of the form $f(x)=\int _{a}^{b} x^{\mu } \sigma (\mu ) \, {\text{d}} \mu $ over $[0,1]$, where $\sigma (\mu )$ is some signed Radon measure, or, more generally, of the form $f(x) = {{\langle \sigma (\mu ), x^\mu \rangle }}$, where $\sigma (\mu )$ is some distribution supported on $[a,b]$, with $0 <a < b< \infty $. One example from this class of functions is $x^{c} (\log{x})^{m}=(-1)^{m} {{\langle \delta ^{(m)}(\mu -c), x^\mu \rangle }}$, where $a\leq c \leq b$ and $m \geq 0$ is an integer. Given the desired accuracy $\varepsilon $ and the values of $a$ and $b$, our method determines a priori a collection of noninteger powers $t_{1}$, $t_{2}$, …, $t_{N}$, 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 $x_{1}$, $x_{2}$, …, $x_{N}$, 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 $\varepsilon $ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}{\varepsilon }})$. We demonstrate the performance of our algorithm with several numerical experiments.
期刊介绍:
The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.