Random Blaschke products
HTML articles powered by AMS MathViewer
- by W. George Cochran PDF
- Trans. Amer. Math. Soc. 322 (1990), 731-755 Request permission
Abstract:
Let $\{ {\theta _n}(\omega )\}$ be a sequence of independent random variables uniformly distributed on $[0,2\pi ]$, and let ${z_n}(\omega ) = {r_n}{e^{i{\theta _n}(\omega )}}$ for a fixed but arbitrary sequence of radii ${r_n}$ satisfying the Blaschke condition $\sum {(1 - {r_n}) < \infty }$. We show that the random Blaschke product with zeros ${z_n}(\omega )$ is almost surely not in the little Bloch space, and we describe necessary conditions and sufficient conditions on the radii ${r_n}$ so that $\{ {z_n}(\omega )\}$ is almost surely an interpolating sequence.References
- J. M. Anderson, Bloch functions: the basic theory, Operators and function theory (Lancaster, 1984) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 153, Reidel, Dordrecht, 1985, pp. 1–17. MR 810441
- J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 361090 J. M. Anderson, J. L. Fernández and A. L. Shields, Inner functions and cyclic vectors in the little Bloch space, preprint.
- Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
- Christopher J. Bishop, Bounded functions in the little Bloch space, Pacific J. Math. 142 (1990), no. 2, 209–225. MR 1042042
- Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
- Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
- Kai Lai Chung, A course in probability theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2001. MR 1796326
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361 G. J. Hungerford, Boundaries of smooth sets and singular sets of Blaschke products in the little Bloch space, preprint from a doctoral thesis, California Institute of Technology, 1988.
- J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseign. Math. (2) 15 (1969), 185–192 (French). MR 245734
- A. G. Naftalevič, On interpolation by functions of bounded characteristic, Vilniaus Valst. Univ. Moksl Darbai. Mat. Fiz. Chem. Moksl Ser. 5 (1956), 5–27 (Russian). MR 0120387
- Emanuel Parzen, Modern probability theory and its applications, A Wiley Publication in Mathematical Statistics, John Wiley & Sons, Inc., New York-London, 1960. MR 0112166
- Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR 521811 —, Blaschke products in ${{\mathbf {B}}_0}$, Linear and Complex Analysis Problem Book, ed. by Havin, Hruščëv, and Nikol ’ skiǐ, Lecture Notes in Math., vol. 1043, Springer-Verlag, 1984, pp. 337-338. MR 85k:46001.
- H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892
- Kenneth Stephenson, Construction of an inner function in the little Bloch space, Trans. Amer. Math. Soc. 308 (1988), no. 2, 713–720. MR 951624, DOI 10.1090/S0002-9947-1988-0951624-3
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 322 (1990), 731-755
- MSC: Primary 30D50; Secondary 30B20, 30C15, 30D55
- DOI: https://doi.org/10.1090/S0002-9947-1990-1022163-8
- MathSciNet review: 1022163