Let $\widehat{f}$ be the Fourier transform of $f$ defined by $$ \widehat{f}(\xi) = \int_{\textbf{R}}e^{2\pi i x \xi}f(x)dx, \quad f\in L^2(\textbf{R}). $$ Define a set $$ E:= \{f\in L^2(\textbf{R}): supp \widehat{f} \subset [-1,1]\}. $$ where supp denotes the support of a function.
Note that if $f\in E$ then $f$ can be extended to an entire function on the plane $\textbf{C}$. Thus if $f=0$ on the interval $[-1,1]$ then $f\equiv 0$ on the line $\textbf{R}$, and of course $f=0$ on $[-2,-2]$. My question arises naturally, namely does there exist a quantitative unique continuation of this kind. Precisely, I want to know:
Question: Is there a constant $C$ independent of $f$ such that \begin{equation}\label{equ-UCP} \int_{[-2,2]}|f|^2dx \leq C\int_{[-1,1]}|f|^2dx \end{equation} hold for all $f\in E$?