Stability Estimates for Truncated Fourier and Laplace Transforms

We prove sharp stability estimates for the Truncated Laplace Transform and Truncated Fourier Transform. The argument combines an approach recently introduced by Alaifari, Pierce and the second author for the truncated Hilbert transform with classical results of Bertero, Gr\"unbaum, Landau, Pollak and Slepian. In particular, we prove there is a universal constant $c >0$ such that for all $f \in L^2(\mathbb{R})$ with compact support in $[-1,1]$ normalized to $\|f\|_{L^2[-1,1]} = 1$ $$\int_{-1}^{1}{|\widehat{f}(\xi)|^2d\xi} \gtrsim \left(c\left\|f_x \right\|_{L^2[-1,1]} \right)^{- c\left\|f_x \right\|_{L^2[-1,1]}}$$ The inequality is sharp in the sense that there is an infinite sequence of orthonormal counterexamples if $c$ is chosen too small. The question whether and to which extent similar inequalities hold for generic families of integral operators remains open.