Zeros of analytic functions with infinitely differentiable boundary values
Author:
James G. Caughran
Journal:
Proc. Amer. Math. Soc. 24 (1970), 700-704
MSC:
Primary 30.67
DOI:
https://doi.org/10.1090/S0002-9939-1970-0252649-8
MathSciNet review:
0252649
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A necessary and sufficient condition is proved that a set of points $\{ {r_n}{e^{i\theta n}}\}$ in the unit disk be the set of zeros of an analytic function with infinitely differentiable boundary values for every choice of $\{ {r_n}\} ,\;0 < {r_n} < 1\;{\text {and}}\;\sum {(1 - {r_n}) < \infty }$
- Arne Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1–13 (French). MR 1370, DOI https://doi.org/10.1007/BF02546325
- Lennart Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345. MR 50011, DOI https://doi.org/10.1007/BF02392289
- Lennart Carleson, A representation formula for the Dirichlet integral, Math. Z. 73 (1960), 190–196. MR 112958, DOI https://doi.org/10.1007/BF01162477
- James G. Caughran, Two results concerning the zeros of functions with finite Dirichlet integral, Canadian J. Math. 21 (1969), 312–316. MR 236396, DOI https://doi.org/10.4153/CJM-1969-033-5
- W. P. Novinger, Holomorphic functions with infinitely differentiable boundary values, Illinois J. Math. 15 (1971), 80–90. MR 269861
- B. A. Taylor and D. L. Williams, Ideals in rings of analytic functions with smooth boundary values, Canadian J. Math. 22 (1970), 1266–1283. MR 273024, DOI https://doi.org/10.4153/CJM-1970-143-x S. Warschawski, Über einen Satz von O. D. Kellogg, Gött. Nach. (1932), 73-86. J. H. Wells, On the zeros of functions with derivatives in ${H^1}$ and ${H^\infty }$, Canad. J. Math, (to appear).
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30.67
Retrieve articles in all journals with MSC: 30.67
Additional Information
Keywords:
Bounded analytic function,
zero set,
boundary zeros,
domain with smooth boundary,
Carleson set,
Blaschke product
Article copyright:
© Copyright 1970
American Mathematical Society