On the Lattice Generated by Hamel Functions

Abstract

We say that $f:\mathbb{R}\to \mathbb{R}$ is LIF if it is linearly independent over $\mathbb{Q}$ as a subset of $\mathbb{R}^2$ and that it is a Hamel function (HF) if it is a Hamel basis of $\mathbb{R}^2$. In this paper we present a discussion on the lattices generated by the classes HF and LIF. We also investigate extensions of partial LIF functions to HF and LIF functions defined on whole $\mathbb{R}