On the Approximation of Singular Functions by Series of Non-integer Powers

Abstract

In this paper, we describe an algorithm for approximating functions of the form $f(x)={\int }_a^b{x}^{\mu }\sigma (\mu )d\mu$ over $[0,1]$, where $\sigma (\mu )$ is some signed Radon measure, or, more generally, of the form $f(x)=<\sigma (\mu ),x^{\mu }>$, where $\sigma (\mu )$ is some distribution supported on $[a,b]$, with $0<a<b<\infty$. One example from this class of functions is $x^c(\log x{)}^m=(-1{)}^m<{\delta }^{(m)}(\mu -c),x^{\mu }>$, where $a\le c\le b$ and $m\ge 0$ is an integer. Given the desired accuracy $\epsilon$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $t_1$, $t_2$, $\dots$, $t_N$, 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 $x_1$, $x_2$, $\dots$, $x_N$, 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 $\epsilon$ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\log \frac{1}{\epsilon })$. We demonstrate the performance of our algorithm with several numerical experiments.