On the approximation of singular functions by series of noninteger powers

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.