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 : l(2) -> l(2) 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 (b) over tilde = (Z) over tilde(a) over tilde 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 (b) over tilde of b then carry over to convergence rates of finite approximations (a) over tilde of a. Spectral convergence is a special case. Some examples are added for illustration. (C) 2016 Elsevier B.V. All rights reserved.