Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

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 $I_1$ be the interval where f is supported, and $I_2$ be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is $\left. {\mathcal {H}}_1 f=g\right| _{I_2}$, where ${\mathcal {H}}_1$ is the FHT that integrates over $I_1$ and gives the result on $I_2$, i.e. ${\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)$. In the case of complete data, $I_1\subset I_2$, 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), $I_1$ is no longer a subset of $I_2$, and the inversion problems becomes severely unstable. By using a differential operator L that commutes with ${\mathcal {H}}_1$, one can obtain the singular value decomposition of ${\mathcal {H}}_1$. Then the rate of decay of singular values of ${\mathcal {H}}_1$ is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals $I_{1,2}$ are possible. The cases when $I_1$ and $I_2$ 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 ${\mathcal {H}}_1^*{\mathcal {H}}_1$ is discrete, and the asymptotics of its eigenvalues $\sigma _n$ as $n\rightarrow \infty$ has been obtained. In this paper we consider the case when the intervals $I_1=(a_1,0)$ and $I_2=(0,a_2)$ are adjacent. Here $a_1 < 0 < a_2$. 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 $U_1$, $U_2$, such that $U_2{\mathcal {H}}_1 U_1^*$ is the multiplication operator with the function $\sigma (\lambda )$, $\lambda \ge (a_1^2+a_2^2)/8$. Here $\lambda$ is the spectral parameter. Then we show that $\sigma (\lambda )\rightarrow 0$ as $\lambda \rightarrow \infty$ 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 $U_1$, $U_2$ as $\lambda \rightarrow \infty$. When the intervals are symmetric, i.e. $-a_1=a_2$, the operators $U_1$, $U_2$ are obtained explicitly in terms of hypergeometric functions.