Functional analytic approach to stability problems in three-dimensional theoretical tomography (Q1822718)

Let \(P_ i\) be the orthogonal projection onto the subspace \(L_ i\) in some Hilbert space, \(i=1,...,m\). In a former note the author proved the following theorem: Suppose that \(L_ i+L_ j\) is closed for all i, j and \(P_ iP_ jP_ k\) is compact for all \(i\neq j\neq k\neq i\), then \(L_ 1+...+L_ m\) is closed. The aim of the paper is to show how this theorem can be used in the study of stability problems in three-dimensional theoretical tomography. First, the author gives a new proof of \textit{J. Boman}'s theorem [Ann. Inst. Fourier 34, No.1, 207-239 (1984; Zbl 0521.46018)] in the case \(p=2:\) Let \(\Omega\) be an open bounded convex subset of \({\mathbb{R}}^ 3\) whose boundary is of class \(C^ 2\). Let \(a_ i\in S^ 2\), \(i=1,...,m\), and let \(1\leq p\leq \infty\). Assume that both principal curvatures of \(\partial \Omega\) are non-zero at every point. Then \(L^ p(\Omega,a_ 1)+...+L^ p(\Omega,a_ m)\) is closed in \(L^ p(\Omega)\). Here \(L^ p(\Omega,a_ i)\) denotes the set of functions in \(L^ p(\Omega)\) constant on almost every line parallel to \(a_ i.\) In a further section more general results are sketched where the sets of parallel lines are replaced by certain sets of curves.