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 interest (ROI). It was proved recently that the interior problem has a unique solution if fa is assumed to be piecewise polynomial on the ROI. In this paper, we tackle the related question of stability. It is well known that lambda tomography allows one to stably recover the locations and values of the jumps of fa inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, fa on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on fa, we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open set U on which fa is the restriction of a real-analytic function, then fa is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result.

中文翻译：

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感兴趣区域内多项式衰减的内部问题的稳定性

在许多实际应用中，希望在不了解感兴趣区域 (ROI) 内开放集上的衰减函数 fa 的情况下解决断层扫描的内部问题。最近证明，如果假设 fa 是 ROI 上的分段多项式，则内部问题具有唯一解。在本文中，我们解决了稳定性的相关问题。众所周知，lambda 断层扫描允许仅从本地数据中稳定地恢复 ROI 内 fa 跳跃的位置和值。因此，我们在这里只考虑 ROI 上的多项式而不是分段多项式 fa 的情况。假设多项式的次数是已知的，连同对 fa 的一些其他相当温和的假设，我们证明了内部问题的稳定性估计。此外，我们证明以下一般唯一性结果。如果存在一个开集 U，其 fa 是实解析函数的限制，则 fa 唯一地由通过 U 的线积分确定。事实证明，两个已知的唯一性定理是这个结果的推论。