**Papers Published**

- Alaifari, R; Pierce, LB; Steinerberger, S,
*Lower bounds for the truncated Hilbert transform*, Arxiv:1311.6845 [Math], vol. 32 no. 1 (November, 2013), pp. 23-56, European Mathematical Publishing House

(last updated on 2019/07/15)**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 $$ \|Hf\|_{L^2(J)} \geq c_1 \exp{\left(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\right)} \| f \|_{L^2(I)} ,$$ 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 $\|f_x\|_{L^1(I)}$.