Numerical computation of the half Laplacian by means of a fast convolution algorithm

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 $\pi$.