On the accumulation of the zeros of a Blaschke product at a boundary point

Protas, David

doi:10.1090/S0002-9939-1972-0294645-2

On the accumulation of the zeros of a Blaschke product at a boundary point
HTML articles powered by AMS MathViewer

by David Protas

Proc. Amer. Math. Soc. 34 (1972), 489-496

DOI: https://doi.org/10.1090/S0002-9939-1972-0294645-2

PDF | Request permission

Abstract:

Let B be a Blaschke product with zeros $\{ {a_n}\}$. The series $\sum {(1 - |{a_n}{|^2})} /|1 - \bar \zeta {a_n}{|^\gamma }$, where $\gamma \geqq 1$ and $|\zeta | = 1$, has been used by P. R. Ahern, D. N. Clark, G. T. Cargo, and others in the study of the boundary behavior of B and various associated functions. In this paper the convergence of this series is shown to be equivalent to a condition on a reproducing kernel for a subspace of the Hardy space ${H^2}$. Some related conditions and corollaries are also given.

References

P. R. Ahern and D. N. Clark, Radial limits and invariant subspaces, Amer. J. Math. 92 (1970), 332–342. MR 262511, DOI 10.2307/2373326
P. R. Ahern and D. N. Clark, On functions orthogonal to invariant subspaces, Acta Math. 124 (1970), 191–204. MR 264385, DOI 10.1007/BF02394571
P. R. Ahern and D. N. Clark, Radial $n\textrm {th}$ derivatives of Blaschke products, Math. Scand. 28 (1971), 189–201. MR 318495, DOI 10.7146/math.scand.a-11015
G. T. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canadian J. Math. 14 (1962), 334–348. MR 136743, DOI 10.4153/CJM-1962-026-2
G. T. Cargo, The segmental variation of Blaschke products, Duke Math. J. 30 (1963), 143–149. MR 145081

The Taylor series

C. N. Linden and H. Somadasa, On tangential limits of Blaschke products, Arch. Math. (Basel) 18 (1967), 416–424. MR 233985, DOI 10.1007/BF01898836
David Protas, Tangential limits of functions orthogonal to invariant subspaces, Trans. Amer. Math. Soc. 166 (1972), 163–172. MR 293100, DOI 10.1090/S0002-9947-1972-0293100-8
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210528