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Revista Matemática Iberoamericana

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Volume 32, Issue 1, 2016, pp. 23–56
DOI: 10.4171/RMI/880

Published online: 2016-02-29

Lower bounds for the truncated Hilbert transform

Rima Alaifari[1], Lillian B. Pierce[2] and Stefan Steinerberger[3]

(1) ETH Zürich, Switzerland
(2) Duke University, Durham, USA
(3) Yale University, New Haven, USA

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 $f \in H^1(I)$, we show that $$ \|Hf\|_{L^2(J)} \geq c_1 \exp{\Big(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\Big)} \| f \|_{L^2(I)} ,$$ for some constants $c_1, c_2 > 0$ depending only on $I, J$. This inequality is sharp, but we conjecture that $\|f_x\|_{L^2(I)}$ can be replaced by $\|f_x\|_{L^1(I)}$.

Keywords: Hilbert transform, truncated data, total variation, lower bound, stability estimate

Alaifari Rima, Pierce Lillian, Steinerberger Stefan: Lower bounds for the truncated Hilbert transform. Rev. Mat. Iberoam. 32 (2016), 23-56. doi: 10.4171/RMI/880