JournalsrmiVol. 32, No. 1pp. 23–56

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
Lower bounds for the truncated Hilbert transform cover

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Abstract

Given two intervals I,JRI, J \subset \mathbb{R}, we ask whether it is possible to reconstruct a real-valued function fL2(I)f \in L^2(I) from knowing its Hilbert transform HfHf on JJ. 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 ff to functions with controlled total variation, reconstruction becomes stable. In particular, for functions fH1(I)f \in H^1(I), we show that

HfL2(J)c1exp(c2fxL2(I)fL2(I))fL2(I),\|Hf\|_{L^2(J)} \geq c_1 \exp{\Big(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\Big)} \| f \|_{L^2(I)} ,

for some constants c1,c2>0c_1, c_2 > 0 depending only on I,JI, J. This inequality is sharp, but we conjecture that fxL2(I)\|f_x\|_{L^2(I)} can be replaced by fxL1(I)\|f_x\|_{L^1(I)}.

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