Periodic Schur functions and slit discs

Abstract

A simply connected domain G is called a slit disc if G = ID minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2 pi. The conformal mapping phi of D onto G, phi(0) = 0, phi'(0) > 0, extends to a continuous function on T mapping it onto partial derivative G. A finite union E of closed non-intersecting arcs e(k) on T is called rational if nu(E)(e(k)) is an element of Q for every k, nu(E)(e(k)) being the harmonic measures of e(k) at infinity for the domain C \ E. A compact E is rational if and only if there is a rational slit disc G such that E = phi(-1) (T). A compact E essentially supports a measure with periodic Verblunsky parameters if and only if E = phi(-1) (T) for a rationally placed G. For any tuple (alpha(1), ... , alpha(g+1)) of positive numbers with Sigma(k) alpha(k) = 1 there is a finite family {e(k)}(k=1)(g+1) of closed non-intersecting arcs e(k) on T such that nu(E) (e(k)) = alpha(k). For any set E = boolean OR(g+1)(k=1) e(k) subset of T and any epsilon > 0 there is a rationally placed compact E* = boolean OR(g+1)(k=1) e(k)* Such that the Lebesgue measure vertical bar E Delta E*vertical bar of the symmetric difference E Delta E* is smaller than epsilon. (C) 2009 Elsevier Inc. All rights reserved.