Orthogonal basises can be constructed in any Hilbert space $L^2(Q)$ in any interval $Q$ by using any countable dense family of function (the Gram-Schmidt recipe), and they are indexed from $1$ to $\infty$. Depending on the Lie group, we have them as special functions:
But how does the basis $\{e^{2πikt}\}_{k=-\infty}^\infty$ with index going to $-\infty$ constructed in $L^2(\Bbb S^1)$? This is given at the beginning of any text as an example of orthonormality, but none of them explains how the basis is constructed. I suspect it's related to the fact that the eigenvalues of the generator of the group $SO(2)$ must be in $\Bbb Z$ due to the restriction of the rotation group in $\Bbb R^2$ ($J|m\rangle=|m\rangle m, m\in\mathbb Z$). The irreducible representation of $SO(2)$ is $U^m(\phi)=e^{-iφm}$ and from the Schur's lemma we can deduce the orthogonality and completeness, which is the Fourier series. However the translation group in the same space does not have this restriction, and in both case I don't see how $-\infty$ comes up.