Math @ Duke

Publications [#320388] of Lillian B. Pierce
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. 2356 [doi]
(last updated on 2018/11/15)
Abstract: Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a realvalued 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 illposed. 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 $\f_x\_{L^1(I)}$.


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