Beurling-Malliavin theory for Toeplitz kernels

We consider the family of Toeplitz operators T_(JS[overbar]^a) acting in the Hardy space H^2 in the upper halfplane; J and S are given meromorphic inner functions, and a is a real parameter. In the case where the argument of S has a power law type behavior on the real line, we compute the critical value c(J, S) = inf{a: ker T_(JS[overbar]^a) ≠ 0} The formula for c(J,S) generalizes the Beurling-Malliavin theorem on the radius of completeness for a system of exponentials

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Evidence against global attention filters selective for absolute bar-orientation in human vision

The finding that an item of type A pops out from an array of distractors of type B typically is taken to support the inference that human vision contains a neural mechanism that is activated by items of type A but not by items of type B. Such a mechanism might be expected to yield a neural image in which items of type A produce high activation and items of type B low (or zero) activation. Access to such a neural image might further be expected to enable accurate estimation of the centroid of an ensemble of items of type A intermixed with to-be-ignored items of type B. Here, it is shown that as the number of items in stimulus displays is increased, performance in estimating the centroids of horizontal (vertical) items amid vertical (horizontal) distractors degrades much more quickly and dramatically than does performance in estimating the centroids of white (black) items among black (white) distractors. Together with previous findings, these results suggest that, although human vision does possess bottom-up neural mechanisms sensitive to abrupt local changes in bar-orientation, and although human vision does possess and utilize top-down global attention filters capable of selecting multiple items of one brightness or of one color from among others, it cannot use a top-down global attention filter capable of selecting multiple bars of a given absolute orientation and filtering bars of the opposite orientation in a centroid task

Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle

This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly supported measures or, more generally, Fourier–Weyl–Titchmarsh transforms associated with selfadjoint differential operators with compact resolvent