Berezin transform and Toeplitz operators on polygonal domains

We consider reflexive Bergman spaces A(p)(Omega) on polygonal domains Omega of the complex plane. With some restrictions to the angles of the boundary of Omega, we show that the boundedness of the Toeplitz operator T-g : A(p)(Omega) -> A(p)(Omega) with a positive symbol g is equivalent to the boundedness of the Berezin transform of g or to g times the area measure being a Carleson measure. The result is also formulated for more general simply connected domains. The main technical tool is a new weighted Forelli-Rudin-type estimate.Peer reviewe

Fourth Moments and Independent Component Analysis

In independent component analysis it is assumed that the components of the observed random vector are linear combinations of latent independent random variables, and the aim is then to find an estimate for a transformation matrix back to these independent components. In the engineering literature, there are several traditional estimation procedures based on the use of fourth moments, such as FOBI (fourth order blind identification), JADE (joint approximate diagonalization of eigenmatrices), and FastICA, but the statistical properties of these estimates are not well known. In this paper various independent component functionals based on the fourth moments are discussed in detail, starting with the corresponding optimization problems, deriving the estimating equations and estimation algorithms, and finding asymptotic statistical properties of the estimates. Comparisons of the asymptotic variances of the estimates in wide independent component models show that in most cases JADE and the symmetric version of FastICA perform better than their competitors.Comment: Published at http://dx.doi.org/10.1214/15-STS520 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org