Integral Equations and Operator Theory

, Volume 87, Issue 4, pp 529–543 | Cite as

Lower Bounds for Truncated Fourier and Laplace Transforms

Article

Abstract

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, Grünbaum, 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 \([-1,1]\) and normalized to \(\Vert f\Vert _{L^2[-1,1]} = 1\)
$$\begin{aligned} \int _{-1}^{1}{|{\widehat{f}}(\xi )|^2d\xi } \gtrsim \left( c\left\| f_x \right\| _{L^2[-1,1]} \right) ^{- c\left\| f_x \right\| _{L^2[-1,1]}} \end{aligned}$$
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.

Keywords

Truncated Laplace transform Truncated Fourier transform Stability estimates Slepian’s ‘happy miracle’ 

Mathematics Subject Classification

Primary 44A15 Secondary 45A05 45Q05 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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