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Inner functions and cyclic vectors in the Bloch space. (English) Zbl 0768.46003
Summary: We construct a singular inner function whose polynomial multiples are dense in the little Bloch space \({\mathcal B}_ 0\). To do this we construct a singular measure on the unit circle with “best possible” control of both the first and second differences.

MSC:
46E15 Banach spaces of continuous, differentiable or analytic functions
30H05 Spaces of bounded analytic functions of one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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[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, K. F. Barth, and D. A. Brannan, Research problems in complex analysis, Bull. London Math. Soc. 9 (1977), no. 2, 129 – 162. · Zbl 0354.30002 · doi:10.1112/blms/9.2.129 · doi.org
[3] 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
[4] S. N. Bernšteĭn, On majorants of finite or quasi-finite growth, Doklady Akad. Nauk SSSR (N.S.) 65 (1949), 117 – 120 (Russian).
[5] Leon Brown and Allen L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), no. 1, 269 – 303. · Zbl 0517.30040
[6] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19 – 42. · Zbl 0301.60035
[7] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. · Zbl 0053.26802
[8] P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247 – 254. · Zbl 0174.37501
[9] Philip Hartman and Richard Kershner, The Structure of Monotone Functions, Amer. J. Math. 59 (1937), no. 4, 809 – 822. · Zbl 0018.01202 · doi:10.2307/2371350 · doi.org
[10] J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseignement Math. (2) 15 (1969), 185 – 192 (French). · Zbl 0175.33902
[11] Boris Korenblum, Cyclic elements in some spaces of analytic functions, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 3, 317 – 318. · Zbl 0473.30031
[12] George Piranian, Allen L. Shields, and James H. Wells, Bounded analytic functions and absolutely continuous measures, Proc. Amer. Math. Soc. 18 (1967), 818 – 826. · Zbl 0188.44904
[13] James W. Roberts, Cyclic inner functions in the Bergman spaces and weak outer functions in \?^\?,0<\?<1, Illinois J. Math. 29 (1985), no. 1, 25 – 38. · Zbl 0562.30041
[14] L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 235 – 277. · Zbl 0152.13202
[15] L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276 – 280. · Zbl 0197.39001
[16] Harold S. Shapiro, Weakly invertible elements in certain function spaces, and generators in \cal\?\(_{1}\), Michigan Math. J. 11 (1964), 161 – 165. · Zbl 0133.37303
[17] Harold S. Shapiro, Monotonic singular functions of high smoothness, Michigan Math. J. 15 (1968), 265 – 275. · Zbl 0165.06904
[18] Joel Shapiro, Cyclic inner functions in Bergman spaces, preprint, 1980 (not for publication).
[19] Allen L. Shields, Cyclic vectors in Banach spaces of analytic functions, Operators and function theory (Lancaster, 1984) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 153, Reidel, Dordrecht, 1985, pp. 315 – 349. · Zbl 0604.46028
[20] Allen L. Shields and David L. Williams, Bounded projections, duality, and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 256 – 279. · Zbl 0367.46053
[21] Allen L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), no. 1, 3 – 25. · Zbl 0508.31001
[22] E. M. Stein and A. Zygmund, On the differentiability of functions, Studia Math. 23 (1963/1964), 247 – 283. · Zbl 0122.30203
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