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}
}
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
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
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 $\Omega \subset \mathbb{R}^2$ be 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
TLDR
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
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
TLDR
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
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
...
1
2
...

References

SHOWING 1-10 OF 28 REFERENCES
Spectral Analysis of the Truncated Hilbert Transform with Overlap
TLDR
The Sturm--Liouville operator is found to commute with H_T, which implies that the spectrum of $H_T^* H-T$ 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
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
TLDR
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
Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals
The purpose of this paper is to examine the mathematical truth in the engineering intuition that there are approximately 2WT independent signals ϕ i of bandwidth W concentrated in an interval ofExpand
The interior Radon transform
The interior Radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured. This problem occurs due toExpand
Finite Hilbert transform with incomplete data: null-space and singular values
Using the Gelfand–Graev formula, the interior problem of tomography reduces to the inversion of the finite Hilbert transform (FHT) from incomplete data. In this paper, we study several aspects ofExpand
Singular value decomposition for the truncated Hilbert transform
Starting from a breakthrough result by Gelfand and Graev, inversion of the Hilbert transform became a very important tool for image reconstruction in tomography. In particular, their result is usefulExpand
Asymptotic Expansions—I
The interest in asymptotic analysis originated from the necessity of searching for approximations to functions close the point(s) of interest. Suppose we have a function f(x) of single real parameterExpand
Stability of the interior problem with polynomial attenuation in the region of interest
In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function fa on an open set within the region of interestExpand
Truncated Hilbert transform and image reconstruction from limited tomographic data
A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection ofExpand
...
1
2
3
...