A convergence theorem for harmonic measures with applications to Taylor series

Gardiner, Stephen J.; Manolaki, Myrto

doi:10.1090/proc/12764

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…So, the result of Fatou and M. Riesz does not apply here. On the other hand, without posing any conditions on the growth of (a n ), a recent result of Gardiner and Manolaki (see [11]) shows that arbitrary functions f holomorphic in D have the following remarkable property:…”

Section: Immediately Impliesmentioning

confidence: 99%

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

Beise

Müller

2016

Math. Z.

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It is known that, generically, Taylor series of functions holomorphic in the unit disc turn out to be universal series outside of the unit disc and in particular on the unit circle. Due to classical and recent results on the boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman spaces the situation is essentially different. In this paper it is shown that in many respects these results are sharp in the sense that universality generically appears on maximal exceptional sets. As a main tool it is proved that the Taylor (backward) shift on certain Bergman spaces is mixing.

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Section: Immediately Impliesmentioning

confidence: 99%

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

Beise

Müller

2016

Math. Z.

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“…(Analogous reasoning for the function f (s) = (2 s − 1) −1 yields a corresponding counterexample for ordinary Dirichlet series.) Theorems 1 and 2 provide analogues for Dirichlet series of results recently established for Taylor series in [12] and [10], respectively. Theorem 4 implies a corresponding result for Taylor series a n z n in the unit disc D with (pure) Ostrowski gaps, which can readily be deduced using the substitution z = e −s (see Corollary 14 in Section 6).…”

Section: General Dirichlet Seriesmentioning

confidence: 77%

“…Proof of Theorem 1. We adapt an argument from [12]. Let f (s) = ∞ n=1 a n e −λns , where f ∈ D(C + ), and let (S m k ), E and F be as in the statement of Theorem 1.…”

Section: Proofmentioning

confidence: 99%

Boundary behaviour of Dirichlet series with applications to universal series

Gardiner

Manolaki

2016

Bull. London Math. Soc.

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This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties. General Dirichlet seriesWe consider a general Dirichlet series of the form f (s) = ∞ n=1 a n e −λns , where (λ n ) is an unbounded strictly increasing sequence of nonnegative real numbers. If this series converges when s = s 0 , then, as is well known, it converges uniformly throughout any angular region of the form {s ∈ C : |arg(s − s 0 )| < π/2 − δ}, where δ ∈ (0, π/2). In particular, f is defined on the half-plane {Re s > Re s 0 } and has a nontangential limit at s 0 . No such conclusion may be drawn if we know merely that some subsequence of the partial sums of the series converges at s = s 0 (see, for example, Bayart [3]). Nevertheless, we show below that there are strong connections between the nontangential boundary behaviour of f and the limiting behaviour of such subsequences. This has implications for the boundary behaviour of universal Dirichlet series and universal Taylor series.A typical point of the complex plane will be written as s = σ + it. We denote by C + the right-hand half-plane {σ > 0}, and by D(C + ) the space of all holomorphic functions on C + which can be represented there as an absolutely convergent Dirichlet series

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“…An interesting question is whether there are (finite) limit functions different from f on parts of T ∩ Ω. A recent result of Gardiner and Manolaki (see [10]) which is based on deep tools from potential theory shows that functions f ∈ H(D) have the following remarkable property.…”

Section: Applicationsmentioning

confidence: 99%

Mixing Taylor shifts and universal Taylor series

Beise¹,

Meyrath²,

Müller

2014

Bulletin of the London Mathematical Society

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It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so‐called universality) and overconvergence in parts of the domain. Our aim is to show how the theory of universal Taylor series can be put into the framework of linear dynamics. This leads to a unified approach to universality and overconvergence and yields new insight into the boundary behaviour of Taylor series.

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A convergence theorem for harmonic measures with applications to Taylor series

Cited by 10 publications

References 15 publications

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

Boundary behaviour of Dirichlet series with applications to universal series

Mixing Taylor shifts and universal Taylor series

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