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

TL;DR

This paper introduces an algorithm for approximating a class of singular functions using series of non-integer powers, with proven accuracy and efficiency, demonstrated through numerical experiments.

Contribution

The paper presents a novel algorithm that determines non-integer power series approximations for singular functions with guaranteed accuracy and logarithmic growth in the number of terms.

Findings

Approximation error is proportional to the desired accuracy psilon.

Number of terms grows as O(log(1/psilon)).

Numerical experiments confirm the effectiveness of the method.

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 $a\leq c \leq b$ and $m \geq 0$ 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 $t_1$, $t_2$, $\ldots$, $t_N$, 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 $\epsilon$ 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.