Operator Theory and Inverse Problems

算子理论和反问题摘要本文的研究将集中在几个反问题上：多变矩问题的唯一性，有限多个规定矩的平面域的最佳逼近；两个平面域的Bergman空间之间的限制算子模的逆谱问题；重整化Riesz变换及其在任意维图像重建中的应用。这项工作将结合算子论、函数论和逼近论的方法。在现代活动的许多领域，需要从部分的，有时是遥远的和间接的数据中重建一个实体。相反的问题解决了这样的具体问题。拟议的研究将涉及几个逆问题，例如在层析成像或地球物理中出现的逆问题。更具体地说，给定一个物体的多个不同角度的X射线，我们将提出一种建设性的、最优的近似或有时重建该物体的方法。

Operator Theory and Inverse Problems Abstract The proposed research will focus on a few inverse problems: the uniqueness in the multivariable moment problem, thebest approximation of planar domains with finitely many prescribedmoments; the inverse spectral problem for the modulus of therestriction operator between the Bergman spaces of two planardomains; a renormalized Riesz transform with applications toimage reconstruction in any dimension. Methods of operatortheory, function theory and approximation theory will be combinedin this work. In many areas of modern activity it is required to reconstructan entity from partial, and sometimes distant and indirect, data.Inverse problems address such specific questions. The proposedresearch will be concerned with a couple of inverse problemsarising for instance in tomography or in geophysics. To bemore specific, given a number of different angle X-rays of a body,we will propose a constructive, optimal way of approximating, or sometimes reconstructing this body.