In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the fractional Laplacian $(-\Delta)^{\alpha/2}$ of a function on $\mathbb{R}$ for $\alpha=1$; this case, commonly referred to as the half Laplacian, is equivalent to the Hilbert transform of the derivative of the function. The main ideas are as follows. Given a twice continuously differentiable bounded function $u\in \mathit{C}_b^2(\mathbb{R})$, we apply the change of variable $x=L\cot(s)$, with $L>0$ and $s\in[0,\pi]$, which maps $\mathbb{R}$ into $[0,\pi]$, and denote $(-\Delta)_s^{1/2}u(x(s)) \equiv (-\Delta)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}(x + i)^k/(1+x^2)^{k/2}$. On a previous work, we considered the case with $k$ even and $\alpha\in(0,2)$, so we focus now on the case with $k$ odd. More precisely, we express $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussian hypergeometric function ${}_2F_1$, and also as a well-conditioned finite sum. Then, we use a fast convolution result, that enable us to compute very efficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$, for extremely large values of $M$. This enables us to approximate $(-\Delta)_s^{1/2}u(x(s))$ in a fast and accurate way, especially when $u(x(s))$ is not periodic of period $\pi$.

翻译：本文提出一种快速且精确的伪谱方法，用于数值逼近定义在实数集 $\mathbb{R}$ 上的函数在 $\alpha=1$ 情形下的分数阶拉普拉斯算子 $(-\Delta)^{\alpha/2}$；这一情形通常称为半拉普拉斯算子，等价于函数导数的希尔伯特变换。主要思路如下：给定一个二次连续可微且有界函数 $u\in \mathit{C}_b^2(\mathbb{R})$，我们应用变量变换 $x=L\cot(s)$，其中 $L>0$、$s\in[0,\pi]$，该变换将 $\mathbb{R}$ 映射至区间 $[0,\pi]$，并记 $(-\Delta)_s^{1/2}u(x(s)) \equiv (-\Delta)^{1/2}u(x)$。因此，通过对 $u(x(s))$ 进行傅里叶级数展开，问题被简化为计算 $(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}(x + i)^k/(1+x^2)^{k/2}$。在前期工作中，我们已考虑 $k$ 为偶数且 $\alpha\in(0,2)$ 的情形，故本文重点关注 $k$ 为奇数的情形。具体而言，我们利用高斯超几何函数 ${}_2F_1$ 将 $k$ 为奇数时的 $(-\Delta)_s^{1/2}e^{iks}$ 表达为解析形式，同时给出良态有限和表达式。接着，我们采用快速卷积算法，使得对于极大数值 $M$ 的情形，能够高效计算 $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$。该方法可快速且精确地逼近 $(-\Delta)_s^{1/2}u(x(s))$，尤其适用于 $u(x(s))$ 不以 $\pi$ 为周期的情形。