In this paper, we describe an algorithm for approximating functions of the form $f(x) = < \sigma(\mu), x^\mu >$ over $[0,1] \subset \mathbb{R}$, 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. We demonstrate the performance of our algorithm with several numerical experiments, and show that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}{\epsilon}})$.

翻译：本文描述了一种用于逼近形如 $f(x) = < \sigma(\mu), x^\mu >$ 在 $[0,1] \subset \mathbb{R}$ 上的函数的算法，其中 $\sigma(\mu)$ 是支撑在 $[a,b]$ 上的某个分布，且 $0 < a < b < \infty$。该类函数的一个示例为 $x^c (\log{x})^m=(-1)^m < \delta^{(m)}(\mu-c), x^\mu >$，其中 $a \leq c \leq b$，且 $m \geq 0$ 为整数。给定所需精度 $\epsilon$ 以及 $a$ 和 $b$ 的取值，我们的方法能先验地确定一组非整数幂 $t_1, t_2, \ldots, t_N$，使得函数可通过形如 $f(x) \approx \sum_{j=1}^N c_j x^{t_j}$ 的级数逼近，同时确定一组配置点 $x_1, x_2, \ldots, x_N$，使得展开系数可通过在这些点上对函数进行配置来求得。我们证明了该方法具有较小的均匀逼近误差，该误差与 $\epsilon$ 乘以若干小常数成正比。通过多个数值实验验证了算法的性能，并表明奇异幂和配置点的数量以 $N = O(\log{\frac{1}{\epsilon}})$ 的规模增长。