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=Zab=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 ZZ only, not on the discretization. Convergence rates of finite approximations b̃ of bb then carry over to convergence rates of finite approximations ã of aa. Spectral convergence is a special case. Some examples are added for illustration.