Measures with zeros in the inverse of their moment matrix

We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure $\mu$ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with $\mu$.Comment: Published in at http://dx.doi.org/10.1214/07-AOP365 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear inverse problem by T-matrix completion. I. Theory

We propose a conceptually new method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential $V$ from external scattering data. Although we seek a local (diagonally-dominated) $V$ as the solution to the posed problem, we allow $V$ to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between $V$ and the T-matrix of the problem, $T$. Here it is important to realize that not every $T$ corresponds to a diagonal $V$ and we, therefore, relax the usual condition of strict diagonality (locality) of $V$. An iterative algorithm is proposed in which we seek $T$ that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential $V$ that is as diagonally-dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.Comment: This is Part I of a paper series containing theory only. Part II contains simulations and is available as arXiv:1505.06777 [math-ph]. Accepted in this form to Phys. Rev.