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Universal Taylor series with maximal cluster sets. (English) Zbl 1186.30003

The authors link the overconvergence properties of certain Taylor series in the unit disk \(\mathbb D\) to the maximality of their cluster sets, so connecting outer wild behaviour with inner wild behaviour.
A function \(f\) holomorphic in \(\mathbb{D}\), where \(f(z)=\sum^\infty_{k=0}\), is called a universal Taylor series if, for every compact subset \(K\) of \(\mathbb{C}\setminus\overline{\mathbb{D}}\) with \(\mathbb{C}\setminus K\) connected, and every function \(h\) continuous on \(K\) and holomorphic in the interior of \(K\), there exists an increasing sequence \((\lambda_n)\) of integers such that the \(\lambda_n\)th partial sum of the power series for \(f\) converges uniformly to \(h\) on \(K\) as \(n\to\infty\). This notion was introduced by V. Nestoridis [Ann. Inst. Fourier 46, No. 5, 1293–1306 (1996; Zbl 0865.30001)].
The authors prove the existence of a dense linear submanifold of holomorphic functions in \(\mathbb{D}\) that are, except for zero, universal Taylor series and simultaneously have maximal cluster sets along many curves tending to the boundary. This complements an earlier result of A. Melas and V. Nestoridis [Adv. Math. 157, No. 2, 138–176 (2001; Zbl 0985.30023)]. It follows that \(\partial\mathbb D\) is a natural boundary for every universal Taylor series, thus proving a conjecture of J.-P. Kahane [J. Anal. Math. 80, 143–182 (2000; Zbl 0961.42001)].
Next, using a method of W. J. Schneider [Approximation and harmonic measure, Aspects of contemporary complex analysis, Proc. instr. Conf., Durham/Engl. 1979, 333–349 (1980; Zbl 0505.30025)], they construct a dense linear manifold of universal Taylor series having, for each boundary point, limit zero along some path tangential to the corresponding radius.
Finally, using an extension of Mergelyan’s Approximation Theorem due to F. Bayart [Mich. Math. J. 53, No. 2, 291–303 (2005; Zbl 1092.46006)], they prove the existence of a closed infinite manifold of holomorphic functions with both outer wild behaviour and inner wild behaviour.

MSC:

30B30 Boundary behavior of power series in one complex variable; over-convergence
30D40 Cluster sets, prime ends, boundary behavior
30E10 Approximation in the complex plane
47B38 Linear operators on function spaces (general)
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