In this paper, we first introduce three series of Legendre rational basis functions on the half/whole line by the diagonalization technique and the matrix decomposition technique. The new basis functions are mutually orthogonal in both $L^2$- and $H^1$-inner products, and lead to diagonal systems for second order problems with constant coefficients. Then we construct efficient space-time spectral methods for parabolic problems in unbounded domains using Legendre rational approximation in space and Legendre-Gauss collocation method in time, which can be implemented in a synchronous parallel fashion. Numerical results demonstrate that the use of simultaneously orthogonal basis functions in space may greatly simplify the implementation of the space-time spectral methods. Using these suggested methods, higher accuracy can also be obtained.

在本文中，我们首先通过对角化技术和矩阵分解技术在半/整线上介绍了三个系列的勒让德有理基函数。新的基函数在两者中相互正交${升}^2$- 和 $H^1$-内积，并导致具有常系数的二阶问题的对角系统。然后，我们使用空间上的勒让德有理逼近和时间上的勒让德-高斯搭配方法，为无界域中的抛物线问题构建了有效的时空谱方法，这些方法可以以同步并行方式实现。数值结果表明，在空间中同时使用正交基函数可以大大简化时空谱方法的实现。使用这些建议的方法，还可以获得更高的准确度。