## Random Blaschke products.(English)Zbl 0709.30031

The paper deals with infinite Blaschke products whose zeros are chosen in a random way. The motivation for the author to study such products are twofold. Firstly, the question asked by D. Sarason: will an infinite Blaschke product whose zeros are chosen randomly (in some sense) be in the little Bloch space? Secondly, the following old result due to A. G. Naftalevich: given a sequence $$\{r_ n\}$$, $$0<r_ n<1$$, satisfying the Blaschke condition $$\sum (1-r_ n)<\infty$$, one can find an interpolating sequence of complex numbers $$z_ n$$ with $$| z_ n| =r_ n$$ for every n. It turns out that Theorem 1 of the paper answers negatively the above question of Sarason.
Theorem 1. Let $$\{r_ n\}$$ be a sequence of numbers such that $$\sum (1- r_ n)<\infty$$ and let $$\{\theta_ n(\omega)\}$$ be a sequence of independent random variables which are uniformly distributed on [0,2$$\pi$$ ]. Then for the Blaschke product B with zeros $$r_ ne^{i\theta_ n(\omega)}$$ one has $\limsup_{n\to \infty}(1-| z_ n|)| B'(z_ n)| =1\quad almost\quad surely.$ The paper also contains a detailed proof of the above mentioned result of A. G. Naftalevich. The methods used in the paper form a combination of analytic and probabilistic tools.
Reviewer: J.Janas

### MSC:

 30D50 Blaschke products, etc. (MSC2000) 30B20 Random power series in one complex variable 46E15 Banach spaces of continuous, differentiable or analytic functions

### Keywords:

Blaschke products; little Bloch space
Full Text:

### References:

 [1] 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. · Zbl 0578.30042 [2] J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12 – 37. · Zbl 0292.30030 [3] J. M. Anderson, J. L. Fernández and A. L. Shields, Inner functions and cyclic vectors in the little Bloch space, preprint. · Zbl 0768.46003 [4] 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. · Zbl 0649.60001 [5] Christopher J. Bishop, Bounded functions in the little Bloch space, Pacific J. Math. 142 (1990), no. 2, 209 – 225. · Zbl 0652.30024 [6] Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921 – 930. · Zbl 0085.06504 [7] Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547 – 559. · Zbl 0112.29702 [8] Kai Lai Chung, A course in probability theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2001. · Zbl 0345.60003 [9] Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. [10] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024 [11] Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74 – 111. · Zbl 0192.48302 [12] 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. [13] J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseignement Math. (2) 15 (1969), 185 – 192 (French). · Zbl 0175.33902 [14] 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). [15] Emanuel Parzen, Modern probability theory and its applications, A Wiley Publication in Mathematical Statistics, John Wiley & Sons, Inc., New York-London, 1960. · Zbl 0089.33701 [16] 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. · Zbl 0398.30027 [17] -, Blaschke products in $${{\mathbf{B}}_0}$$, Linear and Complex Analysis Problem Book, ed. by Havin, Hruščëv, and Nikol ’ skii, Lecture Notes in Math., vol. 1043, Springer-Verlag, 1984, pp. 337-338. MR 85k:46001. [18] H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513 – 532. · Zbl 0112.29701 [19] Kenneth Stephenson, Construction of an inner function in the little Bloch space, Trans. Amer. Math. Soc. 308 (1988), no. 2, 713 – 720. · Zbl 0654.30024 [20] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. · Zbl 0085.05601
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