Periodic Schur functions and slit discs

• Authors: S. Khrushchev
• Venue: Journal of Approximation Theory
• Year: 2009

A simply connected domain G is called a slit disc if G=D 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@p. The conformal mapping @f of D onto G, @f(0)=0, @f^'(0)0, extends to a continuous function on T mapping it onto @?G. A finite union E of closed non-intersecting arcs e"k on T is called rational if @n"E(e"k)@?Q for every k, @n"E(e"k) being the harmonic measures of e"k at ~ for the domain C@?E. A compact E is rational if and only if there is a rational slit disc G such that E=@f^-^1(T). A compact E essentially supports a measure with periodic Verblunsky parameters if and only if E=@f^-^1(T) for a rationally placed G. For any tuple (@a"1,...,@a"g"+"1) of positive numbers with @?"k@a"k=1 there is a finite family {e"k}"k"="1^g^+^1 of closed non-intersecting arcs e"k on T such that @n"E(e"k)=@a"k. For any set E=@?"k"="1^g^+^1e"k@?T and any @e0 there is a rationally placed compact E^*=@?"k"="1^g^+^1e"k^* such that the Lebesgue measure |E@?E^*| of the symmetric difference E@?E^* is smaller than @e.