Lower bounds for the truncated Hilbert transform

Abstract

Abstract. Given two intervals I, J ⊂ R, we ask whether it is possible to reconstruct a real-valued function f ∈ 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 ∈ H 1 (I), we show thatfor some constants c 1 , c 2 > 0 depending only on I, J. This inequality is sharp, but we conjecture that fx L 2 (I) can be replaced by fx L 1 (I) .

Lower bounds for the Dyadic Hilbert transform

“…The counter-part of this strategy is that it relies on a "happy accident" (as termed by Slepian) that does not shed light on the geometric/analytic features at play in the Hilbert transform. Therefore, no hint towards lower bounds for more general Calderón-Zygmund operators, nor towards the conjecture in [3] is obtained through that approach.…”

“…When I ∩ K = ∅, the singular value decomposition of the underlying operator has been given in [10] and this case was further studied by Alaifari, Pierce, and Steinerberger in [3]. It turns out that oscillations of f imply instabilities of the problem.…”

“…It turns out that oscillations of f imply instabilities of the problem. The main result of [3] is that there exist constants c 1 , c 2 depending only on I, K such that, for every f ∈ H 1 (I),…”

Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms

“…This provides an alternative approach to methods, which are used in the literature (c.f. [LP61], [APS13] and the references therein). In particular, we do not use an explicit characterization of the singular values of the Hilbert transform.…”

“…• the exponentσ which appears in the estimate is in general far from the optimal one, which can be obtained from the singular value characterization for the truncated Hilbert transform, c.f. [APS13].…”