关于用非整数幂级数逼近奇异函数

IF 2.3 2区 数学 Q1 MATHEMATICS, APPLIED
Mohan Zhao, Kirill Serkh
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引用次数: 0

摘要

在本文中,我们描述了一种近似形式为$f(x)=\int _{a}^{b} x^{\mu } \sigma (\mu ) \, {\text{d}} \mu $ / $[0,1]$的函数的算法,其中$\sigma (\mu )$是某种带签名的Radon测度,或者更一般地说,形式为$f(x) = {{\langle \sigma (\mu ), x^\mu \rangle }}$,其中$\sigma (\mu )$是$[a,b]$上支持的某种分布,带有$0 <a < b< \infty $。这类函数中的一个例子是$x^{c} (\log{x})^{m}=(-1)^{m} {{\langle \delta ^{(m)}(\mu -c), x^\mu \rangle }}$,其中$a\leq c \leq b$和$m \geq 0$是一个整数。给定期望精度$\varepsilon $和$a$、$b$的值,我们的方法先验地确定了非整数幂$t_{1}$、$t_{2}$、…、$t_{N}$的集合,使函数近似为形式为$f(x)\approx \sum _{j=1}^{N} c_{j} x^{t_{j}}$的级数,并确定了一组搭配点$x_{1}$、$x_{2}$、…、$x_{N}$,通过在这些点上搭配函数可以找到展开系数。我们证明了我们的方法有一个小的均匀近似误差,它与$\varepsilon $乘以一些小常数成正比,并且奇异幂和并置点的数量随着$N=O(\log{\frac{1}{\varepsilon }})$的增长而增长。通过几个数值实验验证了算法的性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.
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来源期刊
IMA Journal of Numerical Analysis
IMA Journal of Numerical Analysis 数学-应用数学
CiteScore
5.30
自引率
4.80%
发文量
79
审稿时长
6-12 weeks
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.
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