# Lower bounds for the truncated Hilbert transform

@article{Alaifari2013LowerBF, title={Lower bounds for the truncated Hilbert transform}, author={Rima Alaifari and L. B. Pierce and Stefan Steinerberger}, journal={arXiv: Classical Analysis and ODEs}, year={2013} }

Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a unique answer (the nullspace is trivial) but is severely ill-posed. We isolate the difficulty and show that by restricting $f$ to functions with controlled total variation, reconstruction becomes stable. In particular, for functions $f \in H^1(I… Expand

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