Consider an orthonormal basis $(\varphi\_n)\_{n \in \mathbb N}$ of $L^2(\mathbb R).$


We consider the functionals $\Phi\_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi\_n, f \varphi\_{n+1} \right\rangle$$


for any $n \in \mathbb N$ where $C^b(\mathbb R)$ are the continuous and bounded functions on $\mathbb R.


I would like to know: 


Consider the union $X:=\bigcup\_{n \in \mathbb N} \operatorname{ker}(\Phi\_n).$ 
Is the complement of this set $X$ non-empty. Is the complement perhaps even dense in $C^b(\mathbb R)$ or does it depend on the basis? 



Do functions exist and are they dense? Or does it depend on the basis?
The complement of $X$ may be empty.


Consider a basis such that $\mathrm{supp}\,\varphi\_k \cap \mathrm{supp}\,\varphi\_{k+1}=\emptyset$ for some $k$ (for example, consisting of Walsh functions on each interval $[n, n+1]$). Then obviously $\mathrm{Ker}\,\Phi\_k = C^b(\mathbb{R})$.


