中文摘要：

首先，建立了晶格Fourier分析的一般理论，并具体研究了六边形区域上周期函数的数值逼近．在此基础上，提出了六边形区域上的椭圆型偏微分方程的周期问题求解的六边形Fourier谱方法，设计了相应谱格式快速实现算法，建立了Fourier谱方法的稳定性与收敛性理论．同方形区域上的经典Fourier谱方法一样，六边形Fourier谱方法可以充分利用快速Fourier变换，并具备了“无穷阶”的谱收敛速度．

英文摘要：

In this paper, a general theory of the lattice Fourier analysis is first established. As a concrete application, numerical approximations to periodic functions on the hexagon are studied. The hexagonal spectral methods are then pro-posed for solving the elliptic partial differential equation on the hexagon. The corresponding fast implementations, the stability, and the convergence analysis are also given in detail. Just as the classic tensorial Fourier spectral methods, the hexagonal Fourier spectral methods take the advantage of the fast Fourier transform, and share the so called ＂infinite order＂ of convergence of the spectral methods.