Lower bounds for the truncated Hilbert transform
Lower bounds for the truncated Hilbert transform
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DOI：
10.4171/rmi/880
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发表时间：
2016-01-01
影响因子：
1.2
通讯作者：
Steinerberger, Stefan
中科院分区：
文献类型：
--
作者：
Alaifari, Rima;Pierce, Lillian B.;Steinerberger, Stefan
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Given two intervals I, J subset of R, we ask whether it is possible to reconstruct a real-valued function f is an element of 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 is an element of H-1(I), we show thatparallel to Hf parallel to(L2(J)) >= c(1) exp(-c(2) parallel to f(x)parallel to(L2(I))/parallel to f parallel to(L2(I)))parallel to f parallel to(L2(I)),for some constants c(1), c(2) > 0 depending only on I, J. This inequality is sharp, but we conjecture that parallel to f(x)parallel to(L2(I)) can be replaced by parallel to f(x)parallel to(L1(I)).