Approximation properties of the $q$-sine bases
Provides theoretical guidance for choosing basis functions in spectral methods for nonlinear boundary value problems, but the results are incremental and domain-specific.
The paper studies approximation properties of q-sine bases for solving p-Poisson boundary value problems and related parabolic equations, determining optimal q values for best approximation.
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.