Riesz basis of Paley-Wiener space. Let us consider the Paley-Wiener space:
$$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$
Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real numbers.
The sequence $\{\operatorname{sinc}(t-\lambda_n)\}_{n\in\mathbb Z}$ is a Riesz basis for $PW_\pi$ if $|\lambda_n-n|\leqq L< 1/4$ (Kadec's $1/4$ Theorem for exponential bases).
I wonder if $\{\operatorname{sinc}(t-\lambda_n)\}_{n\in\mathbb N}$ is still a Riesz basis for $PW_\pi$. I think that this is not true, because one-sided sequences could not generate a Riesz basis in the whole space $PW_\pi$. How can prove (or disprove) it?
 A: Consider the Paley-Wiener space
$$ PW_{\frac{1}{2}} := \{ f \in L^2 (\mathbb{R}) \; | \; \text{supp} \;\hat{f} \subseteq [-\frac{1}{2}, \frac{1}{2}] \} $$
Then the collection $\{\text{sinc} (x - n) \}_{n \in \mathbb{Z}}$ is an orthonormal basis for $PW_{\frac{1}{2}}$. Recall now that a Riesz basis for a Hilbert space $\mathcal{H}$ is a collection $\{U e_k \}_{k \in I}$, where $U : \mathcal{H} \to \mathcal{H}$ is a topological isomorphism and $\{e_k\}_{k \in I}$ is an orthonormal basis for $\mathcal{H}$.
Clearly, the collection $\{\text{sinc} (x - n) \}_{n \in \mathbb{Z}}$ forms a Riesz basis for $PW_{\frac{1}{2}}$, but the collection $\{\text{sinc} (x - n) \}_{n \in \mathbb{N}}$ with the index set $\mathbb{N}$ is not. 
A: By a theorem of Landau, it is well-known that if $\{ \text{sinc}(x-\lambda) \}_{\lambda\in\Lambda}$ is a Riesz basis in $PW_{a}$, then $\Lambda$ must have uniform density in $\mathbb{R}$, which equals exactly $2a$.

H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire
  functions, Acta Mathematica 117, 37-52 (1967).

Clearly, any set $\Lambda$ which is contained in a half-line does not satisfy this property.
