Lower bounds for the truncated Hilbert transform
Rima Alaifari
ETH Zürich, SwitzerlandLillian B. Pierce
Duke University, Durham, USAStefan Steinerberger
Yale University, New Haven, USA
![Lower bounds for the truncated Hilbert transform cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-32-issue-1.png&w=3840&q=90)
Abstract
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
DOI 10.4171/RMI/880