Diagonalization Of The Finite Hilbert Transform On Two Adjacent Intervals

Keywords

Diagonalization; Finite Hilbert transform; Interior problem of tomography; Titchmarsh–Weyl theory

Abstract

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let I1 be the interval where f is supported, and I2 be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is H1f=g|I2, where H1 is the FHT that integrates over I1 and gives the result on I2, i.e. H1: L2(I1) → L2(I2). In the case of complete data, I1⊂ I2, and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), I1 is no longer a subset of I2, and the inversion problems becomes severely unstable. By using a differential operator L that commutes with H1, one can obtain the singular value decomposition of H1. Then the rate of decay of singular values of H1 is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals I1 , 2 are possible. The cases when I1 and I2 are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator H1∗H1 is discrete, and the asymptotics of its eigenvalues σn as n→ ∞ has been obtained. In this paper we consider the case when the intervals I1= (a1, 0 ) and I2= (0 , a2) are adjacent. Here a1< 0 < a2. Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations U1, U2, such that U2H1U1∗ is the multiplication operator with the function σ(λ) , λ≥(a12+a22)/8. Here λ is the spectral parameter. Then we show that σ(λ) → 0 as λ→ ∞ exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators U1, U2 as λ→ ∞. When the intervals are symmetric, i.e. - a1= a2, the operators U1, U2 are obtained explicitly in terms of hypergeometric functions.

Publication Date

12-1-2016

Publication Title

Journal of Fourier Analysis and Applications

Volume

22

Issue

6

Number of Pages

1356-1380

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00041-016-9458-x

Socpus ID

84957597163 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84957597163

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