Lower bounds for the truncated Hilbert transform

  • Rima Alaifari

    ETH Zürich, Switzerland
  • Lillian B. Pierce

    Duke University, Durham, USA
  • Stefan Steinerberger

    Yale University, New Haven, USA

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