On the Approximation of Singular Functions by Series of Non-integer Powers
Mohan Zhao, Kirill Serkh
TL;DR
The paper tackles the challenge of approximating endpoint-singular functions on $[0,1]$ that admit integral representations $f(x)=\int_{a}^{b} x^{\mu} \sigma(\mu) \, d\mu$ by sparse expansions in non-integer powers $x^{t_j}$ with coefficients found via collocation, achieving a uniform error of $O(\epsilon)$ with the number of terms $N=O(\log\frac{1}{\epsilon})$. The authors construct the basis from the singular value decomposition of the truncated Laplace transform, selecting collocation points and exponents from the roots of the leading singular functions, and solve a Vandermonde-like system to obtain the coefficients. They prove and demonstrate that the approximation converges exponentially in $N$, extends to distributions, and remains stable under TSVD perturbations, with the coefficients recoverable from a small, well-conditioned system. The approach offers a priori, geometry- and accuracy-driven basis design that does not require detailed a priori knowledge of singularity types, and it can be integrated to enhance finite element and integral equation methods for PDEs with corner or endpoint singularities. Practically, the method yields a compact, accurate representation suitable for PDE solvers on nonsmooth domains and complex geometries, while providing rigorous error bounds and stable coefficient computation.
Abstract
In this paper, we describe an algorithm for approximating functions of the form $f(x)=\int_{a}^{b} x^μ σ(μ) \, d μ$ over $[0,1]$, where $σ(μ)$ is some signed Radon measure, or, more generally, of the form $f(x) = <σ(μ),\, x^μ>$, where $σ(μ)$ is some distribution supported on $[a,b]$, with $0 <a < b < \infty$. One example from this class of functions is $x^c (\log{x})^m=(-1)^m <δ^{(m)}(μ-c), \, x^μ>$, where $a\leq c \leq b$ and $m \geq 0$ is an integer. Given the desired accuracy $ε$ 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 $ε$ multiplied by some small constants, and that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}ε})$. We demonstrate the performance of our algorithm with several numerical experiments.