Cluster sets of interpolating Blaschke products
In this very interesting paper the authors study the radial cluster sets of Blaschke products. \textit{C. L. Belna, P. Colwell} and \textit{G. Piranian} [Proc. Am. Math. Soc. 93, 267--271 (1985; Zbl 0582.30022)] proved that if \(\{ a_n : n=1, 2,\dots \} \) is a sequence of points in the unit circle and \(\{ K_n : n=1, 2,\dots\} \) is a sequence of nonempty, closed, connected sets in the closed unit disk, then some Blaschke product has the radial cluster set \(K_ n\) at \(a_n\), for every \(n\). The authors raise the following question: Is it possible to assert that there exists an interpolating Blaschke product with that property? They do present a number of surprising answers to this question using results of \textit{K.~Hoffman} [Ann. Math. (2) 86, 74--111 (1967; Zbl 0192.48302)]. Actually, the authors obtain results for the so-called thin products. We recall that a Blaschke product with zero sequence \(\{ a_n\} \) is called thin if \(\lim_{n}(1-| a_n| ^2)| B^\prime (a_n)| =1\). Among others, the authors prove the following results: (i) If \(B\) is a thin product with positive zeros then the radial cluster set of \(B\) at the point \(z=1\) is the interval \([-1,1]\). (ii) There does not exist an interpolating Blaschke product with \([0,1]\) as radial cluster set at \(1\). (iii) There exist thin Blaschke products whose radial cluster set at \(1\) is one of the following: the closed unit disk, an arbitrary closed arc on the unit circle. (iv) For any interval of the form \([x,y]\), where \(xy\neq 0\) and \(-1\leq x<y\leq 1\), there exists an interpolating Blaschke product whose radial cluster set at \(1\) is \([x,y]\).
scientific article; zbMATH DE number 5010215
scientific article