# Lower bounds for the truncated Hilbert transform

@article{Alaifari2013LowerBF,
title={Lower bounds for the truncated Hilbert transform},
author={Rima Alaifari and L. B. Pierce and Stefan Steinerberger},
journal={arXiv: Classical Analysis and ODEs},
year={2013}
}
• Published 2013
• Mathematics
• arXiv: Classical Analysis and ODEs
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 $f \in H^1(I… Expand #### Figures from this paper Lower bounds for the dyadic Hilbert transform • Mathematics • 2016 In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form$\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)\left\Vert f\right\Vert_{L^2(I)}$where$I$and$K$are twoExpand Lower Bounds for Truncated Fourier and Laplace Transforms • Mathematics • 2017 We prove sharp stability estimates for the Truncated Laplace Transform and Truncated Fourier Transform. The argument combines an approach recently introduced by Alaifari, Pierce and the second authorExpand Quantitative projections in the Sturm Oscillation Theorem There is$c_{} > 0$such that for all$f \in C[0,\pi]$with at most$d-1$roots inside$(0,\pi)$\sum_{1 \leq n \leq d}{ \left| \left\langle f, \sin\left( n x\right) \right\rangle \right|} \geqExpand Topological Bounds for Fourier Coefficients and Applications to Torsion Let$\Omega \subset \mathbb{R}^2be a bounded convex domain in the plane and consider \begin{align*} -\Delta u &=1 \qquad \mbox{in}~\Omega \\ u &= 0 \qquad \mbox{on}~\partial \Omega. \end{align*}Expand A Remark on the Arcsine Distribution and the Hilbert transform • Mathematics, Medicine • The journal of fourier analysis and applications • 2019 A localized Parseval-type identity is proved that seems to be new: if f(x)(1-x^2)1/4∈L2(-1,1) and its Hilbert transform Hf vanishes on (-1, 1), then the function f is a multiple of the arcsine distribution. Expand Stability estimates for the regularized inversion of the truncated Hilbert transform • Mathematics • 2015 In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering aExpand Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realizeExpand Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation • Computer Science, Mathematics • J. Approx. Theory • 2021 A best approximation problem under constraint in L2 and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of m is formulated and constructively solved. Expand On two methods for quantitative unique continuation results for some nonlocal operators • Mathematics, Physics • 2020 Abstract In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding theExpand Hilbert transforms and the equidistribution of zeros of polynomials We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work ofExpand #### References SHOWING 1-10 OF 28 REFERENCES Spectral Analysis of the Truncated Hilbert Transform with Overlap • Computer Science, Mathematics • SIAM J. Math. Anal. • 2014 The Sturm--Liouville operator is found to commute with H_T, which implies that the spectrum ofH_T^* H-T$is discrete, and the singular value decomposition of the operator is expressed in terms of the solutions to the Sturm-Liouvil... Expand Stability estimates for the regularized inversion of the truncated Hilbert transform • Mathematics • 2015 In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering aExpand Asymptotic Analysis of the SVD for the Truncated Hilbert Transform with Overlap • Mathematics, Computer Science • SIAM J. Math. Anal. • 2015 This paper exploits the property that H_T commutes with a second-order differential operator$L_S$and the global asymptotic behavior of its eigenfunctions to find the asymPTotics of the singular values and singular functions of$H_T\$. Expand
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