Approximation properties of the $q$-sine bases

TL;DR

This paper investigates the approximation capabilities of $q$-sine bases, aiming to optimize their use in spectral methods for $p$-Poisson problems and related evolution equations.

Contribution

It analyzes the approximation properties of $q$-sine bases and their duals, providing insights for their optimal application in spectral methods for nonlinear PDEs.

Findings

Identification of optimal $q$ values for best approximation

Establishment of non-orthogonal spectral methods for $p$-Poisson

Analysis of $q$-sine bases' approximation properties

Abstract

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.