The contention of this paper is that a spectral method for time-dependent PDEs is basically no more than a choice of an orthonormal basis of the underlying Hilbert space. This choice is governed by a long list of considerations: stability, speed of convergence, geometric numerical integration, fast approximation and efficient linear algebra. We subject different choices of orthonormal bases, focussing on the real line, to these considerations. While nothing is likely to improve upon a Fourier basis in the presence of periodic boundary conditions, the situation is considerably more interesting in other settings. We introduce two kinds of orthonormal bases, T-systems and W-systems, and investigate in detail their features. T-systems are designed to work with Cauchy boundary conditions, while W-systems are suited to zero Dirichlet boundary conditions.

翻译：本文认为，时变偏微分方程的谱方法本质上不过是对底层希尔伯特空间中标准正交基的选择。这种选择受到一系列因素的制约：稳定性、收敛速度、几何数值积分、快速逼近和高效线性代数。我们针对实直线上的不同标准正交基选择，对这些因素进行了考量。尽管在周期性边界条件下，傅里叶基几乎无可替代，但在其他情况下情况则更加有趣。我们引入两类标准正交基——T系统和W系统，并详细研究了它们的特征。T系统专为柯西边界条件设计，而W系统则适用于零狄利克雷边界条件。