  For $q>12/11$ the eigenfunctions of the non-linear eigenvalue problem
associated to the one-dimensional $q$-Laplacian are known to form a Riesz basis
of $L^2(0,1)$. We examine in this paper the approximation properties of this
family of functions and its dual, in order to establish non-orthogonal spectral
methods for the $p$-Poisson boundary value problem and its corresponding
parabolic time evolution initial value problem. The principal objective of our
analysis is the determination of optimal values of $q$ for which the best
approximation is achieved for a given $p$ problem.
