Completeness of $\delta_n = e_n + e_{n+1}$ with $e_n$ a complex exponential

Question

Let $e_n(x) = e^{2 \pi i n x}$. It is well-known that $\{ e_n | n \in \mathbb Z \}$ is an orthonormal basis for $L^2[-1/2,1/2]$. In particular, $\{ e_n | n \in \mathbb Z\}$ is complete in $L^2[-1/2,1/2]$. Now let $\delta_n = e_n + e_{n+1}$. I was wondering if the system $\{ \delta_n | n \in \mathbb Z \}$ is complete in $L^2[-1/2,1/2]$ as well. Or more general: let $$\delta_n = \sum e_{k(n)} $$ be a finite non-trivial linear combiniation of $e_{k(n)}$. Is the system $\{ \delta_n : n \in \mathbb Z \}$ complete in $L^2[-1/2,1/2]$?

Answer 1

In general let $\{e_n\}_{n=-\infty}^\infty$ be an orthonormal basis of an infinite dimensional Hilbert space. Then the set $\{e_n+e_{n+1}\}$ is linearly dense. Indeed, it suffices to show that there is no nonzero element $x$ such that $x\perp e_n+e_{n+1}$ for $n\in \mathbb{Z}.$ By the Parseval identity we have $$\|x\|^2=\sum_{n=-\infty}^\infty |\langle x,e_n\rangle |^2$$ If $x\perp e_n+e_{n+1}$ then $$\langle x,e_n\rangle=-\langle x,e_{n+1}\rangle$$ Hence $$|\langle x,e_n\rangle|=|\langle x,e_{n+1}\rangle|$$ which shows that the sequence $|\langle x,e_n\rangle|$ is constant. As it is square summable, then $\langle x,e_n\rangle=0$ for any $n,$ i.e. $x=0.$

Alternatively it can be shown that $$\sum_{k=0}^n (-1)^k\left (1-{k\over n}\right)[e_k+e_{k+1}]$$ approximate $e_0.$ Simlarly we can approximate any element $e_m.$

The statement can be generalized. For example let $$\delta_n=e_n+e_{n+1}+\ldots +\delta_{n+k-1}$$ for a fixed $k.$ Assume $x\perp \delta_n$ for any $n.$ Then the sequence $a_n:=\langle x,e_n\rangle$ satisfies $$a_n+a_{n+1}+\ldots +a_{n+k-1}=0,\quad n\in \mathbb{Z}\quad (*)$$ Therefore $$a_{n+1}+a_{n+2}+\ldots +a_{n+k}=0,\quad n\in \mathbb{Z}\quad (**)$$ Subtracting $(**)$ and $(*)$ implies $a_n=a_{n+k},$ i.e. the sequence $a_n$ is periodic with period $k.$ Since $a_n$ is square summable we get $a_n=0$ for all $n,$ hence $x=0.$