Lower bounds for the truncated Hilbert transform

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

29 pages, 4 figures