- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:48
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
32
2016
1
Lower bounds for the truncated Hilbert transform
Rima
Alaifari
ETH Zürich, ZÜRICH, SWITZERLAND
Lillian
Pierce
Duke University, DURHAM, UNITED STATES
Stefan
Steinerberger
Yale University, NEW HAVEN, UNITED STATES
Hilbert transform, truncated data, total variation, lower bound, stability estimate
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 $\|f_x\|_{L^1(I)}$.
Integral transforms, operational calculus
Integral equations
23
56
10.4171/RMI/880
http://www.ems-ph.org/doi/10.4171/RMI/880