I have a short question which first requires some slightly elaborate definitions: Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, g_n\rangle=\delta_{m,n}$ and for every $x\in\mathcal{H}$; $x=\sum_n \langle x, g_n\rangle e_n=\sum_n \langle x, e_n\rangle g_n$. In particular, let $\mathcal{H}=L^2[-\pi,\pi]$, so that its dual space under the Fourier transform is the Paley-wiener space $PW_{\pi}$, and let $(e_n)_{n\in\mathbb{Z}}=(e^{i\lambda_n x})_{n\in\mathbb{Z}}$ for some adequate sequence of real numbers $(\lambda_n)$ (e.g. s.t. $\sup_n |\lambda_n-n|<\frac{1}{4}$). Furthermore, define the sequence of functions $(\Phi_n(x)):=(\mathcal{F}^{-1}[g_n])$, which is clearly a Riesz-basis for $PW_{\pi}$. Now to the particular question: The functions $\Phi_n(x)$ clearly has zeros at every $(\lambda_k)_{k\neq n}$ (consider the representation $\Phi_n(\lambda_k)=\int g_n(\xi) e^{i \lambda_k \xi}\mathrm{d}\xi$ and use the fact that $g_n$ and $e_n$ are biorthogonal), but is there some specific reason why $\Phi_n(x)$ cannot have any other zeros than these on the real line?

Thank's for your attention.


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