I want to use a theorem which states that if $\{\phi_n\}_1^{\infty}$ is an orthonormal basis of $L^2(a,b)$ then for all $f\in L^2(a,b)$ we have $\parallel f \parallel^2 = \sum_1^\infty|\langle f,\phi_n\rangle|^2$. The problem is that my ON basis is $\{\phi_n\}_{-\infty}^{\infty}$.
However, I am thinking that if I define a new set $\{\Phi_n\}_1^{\infty}$, where $\Phi_1 = \phi_0$ and $\Phi_n = \phi_{(-1)^n\lfloor\frac{n}{2}\rfloor}$ for $n\geq 2$, then the two sets contain the same elements, so $\{\Phi_n\}_1^{\infty}$ is also an ON basis of $L^2(a,b)$. In this way I could use the theorem anyway.
In general, is there something that could possibly go wrong with such a "re-indexing"?