# 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

A subscription is required to access this article.

## Abstract

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

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 $\|f_x\|_{L^1(I)}$.

## 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