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

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.