Convergence analysis of general spectral methods

If a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system b = Za with a map Z : l2 → l2 that has a continuous inverse, this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system b̃ = Z̃ã is uniquely solvable and is unconditionally stable, i.e. the stability can be made to depend on Z only, not on the discretization. Convergence rates of finite approximations b̃ of b then carry over to convergence rates of finite approximations ã of a. Spectral convergence is a special case. Some examples are added for illustration.