Boundary zero sets of $A^\infty$ functions satisfying growth conditions
HTML articles powered by AMS MathViewer
- by B. A. Taylor and D. L. Williams
- Proc. Amer. Math. Soc. 35 (1972), 155-160
- DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
- PDF | Request permission
Abstract:
Let A denote the algebra of functions analytic in the open unit disc D and continuous in D, and let \[ {A^\infty } = \{ f \in A:{f^{(n)}} \in A,n = 0,1,2, \cdots \} .\] For $f \in A$ denote the set of zeros of f in D by ${Z^0}(f)$, and for $f \in {A^\infty }$ let ${Z^\infty }(f) = \bigcap {_{n = 0}^\infty Z{^0}({f^{(n)}})}$. We study the boundary zero sets of ${A^\infty }$ functions F satisfying, for some sequence $\{ {M_n}\}$ and some $B > 0$, \begin{equation}\tag {$1$}|{F^{(n)}}(z)| \leqq n!{B^n}{M_n},\quad z \in \bar D,n = 0,1,2, \cdots .\end{equation} In particular, when ${M_n} = \exp ({n^p}),p > 1$, it is shown that for E, a proper closed subset of $\partial D$, there exists $F \in {A^\infty }$ satisfying (1) and with ${Z^0}(F) = {Z^\infty }(F) = E$ if and only if $\smallint _{ - \pi }^\pi |\log \rho ({e^{i\theta }},E){|^q}D\theta < + \infty$. Here $\rho (z,E)$ is the distance from z to E and $(1/p) + (1/q) = 1$.References
- Lennart Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325β345. MR 50011, DOI 10.1007/BF02392289 A. Chollet, Ensembles de zΓ©ros de fonctions analytiques dans le disque, appartenent Γ une classe de Gevrey sur le bord, C. R. Acad. Sci. Paris SΓ©r. A-B 269 (1969), A447-A449. MR 41 #5627.
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- B. I. Korenbljum, Quasianalytic classes of functions in a circle, Dokl. Akad. Nauk SSSR 164 (1965), 36β39 (Russian). MR 0212199
- W. P. Novinger, Holomorphic functions with infinitely differentiable boundary values, Illinois J. Math. 15 (1971), 80β90. MR 269861, DOI 10.1215/ijm/1256052822
- B. A. Taylor and D. L. Williams, Zeros of Lipschitz functions analytic in the unit disc, Michigan Math. J. 18 (1971), 129β139. MR 283176, DOI 10.1307/mmj/1029000636
- B. A. Taylor and D. L. Williams, Ideals in rings of analytic functions with smooth boundary values, Canadian J. Math. 22 (1970), 1266β1283. MR 273024, DOI 10.4153/CJM-1970-143-x
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 155-160
- MSC: Primary 30A72
- DOI: https://doi.org/10.1090/S0002-9939-1972-0310253-9
- MathSciNet review: 0310253