2013/11/26 by Rima Alaifari, Lillian B. Pierce, Alaifari, Rima +3
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA
paper · pdf · doi:10.48550/arxiv.1311.6845
29 pages, 4 figures
arxiv created 2015/04/30 · arxiv updated 2015/05/01
Given two intervals I, J ⊂ ℝ, we ask whether it is possible to reconstruct a real-valued function f ∈ L2(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 ∈ H1(I), we show that ‖Hf‖L2(J) ≥ c1 exp(-c2 \frac‖fx‖L2(I)‖f‖L2(I)) ‖ f ‖L2(I) , for some constants c1, c2 > 0 depending only on I, J. This inequality is sharp, but we conjecture that ‖fx‖L2(I) can be replaced by ‖fx‖L1(I).