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Given two intervals , we ask whether it is possible to reconstruct a real-valued function from knowing its Hilbert transform on . 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 to functions with controlled total variation, reconstruction becomes stable. In particular, for functions , we show that
for some constants depending only on . This inequality is sharp, but we conjecture that can be replaced by .
Cite this article
Rima Alaifari, Lillian B. Pierce, Stefan Steinerberger, Lower bounds for the truncated Hilbert transform. Rev. Mat. Iberoam. 32 (2016), no. 1, pp. 23–56