We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\boldsymbol\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

翻译：我们提出了一种基于半离散傅里叶变换的新颖且简单的谱方法，用于离散化分数阶拉普拉斯算子 $(-\Delta)^\frac{\alpha}{2}$。我们提供了数值分析和实验来研究其性能。我们的方法在离散层面上具有与分数阶拉普拉斯算子 $(-\Delta)^\frac{\alpha}{2}$ 相同的符号 $|\boldsymbol\xi|^\alpha$，因此可被视为分数阶拉普拉斯算子的精确离散模拟。这一独特特性使我们的方法有别于其他现有的分数阶拉普拉斯算子数值方法。需注意，我们的方法不同于文献中通常局限于周期性边界条件的傅里叶伪谱方法（参见备注 \ref{remark0}）。数值分析表明，我们的方法可以达到谱精度。我们分析了该方法在求解分数阶泊松方程时的稳定性和收敛性。我们的格式产生一个多层托普利茨刚度矩阵，因此可以开发快速算法以实现高效的矩阵-向量乘法。其计算复杂度为 ${\mathcal O}(2N\log(2N))$，内存存储量为 ${\mathcal O}(N)$，其中 $N$ 为总点数。大量的数值实验验证了我们的分析结果，并证明了该方法在求解各类问题中的有效性。