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 $[0,1] \subset \mathbb{R}$, where $0<a<b<\infty$ and $\sigma(\mu)$ is some signed Radon measure over $[a,b]$ or some distribution supported on $[a,b]$. Given the desired accuracy $\epsilon$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $\{t_j\}_{j=1}^N$, so that the functions are approximated by series of the form $f(x)\approx \sum_{j=1}^N c_j x^{t_j}$, where the expansion coefficients can be found by solving a square, low-dimensional Vandermonde-like linear system using the collocation points $\{x_j\}_{j=1}^N$, also determined a priori by $\epsilon$ and the values of $a$ and $b$. 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：本文描述了一种逼近形如 $f(x)=\int_{a}^{b} x^{\mu} \sigma(\mu) \, d \mu$ 的函数的算法，其定义域为 $[0,1] \subset \mathbb{R}$，其中 $0<a<b<\infty$，$\sigma(\mu)$ 是 $[a,b]$ 上的某个带符号Radon测度或支撑在 $[a,b]$ 上的某分布。给定所需精度 $\epsilon$ 以及 $a$ 和 $b$ 的值，我们的方法先验地确定一组非整数幂 $\{t_j\}_{j=1}^N$，从而将函数近似为形如 $f(x)\approx \sum_{j=1}^N c_j x^{t_j}$ 的级数形式。其中展开系数可通过求解一个同样由 $\epsilon$ 及 $a$、$b$ 先验确定的配置点 $\{x_j\}_{j=1}^N$ 所构成的方形低维类Vandermonde线性方程组获得。我们证明该方法具有与 $\epsilon$ 乘以若干小常数成比例的一致小逼近误差。通过多个数值实验展示了算法的性能，并指出奇异幂与配置点的数量增长满足 $N=O(\log{\frac{1}{\epsilon}})$。