Universal Taylor series, conformal mappings and boundary behaviour

Gardiner, Stephen J.

doi:10.5802/aif.2849

Cited by 14 publications

(12 citation statements)

References 18 publications

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“…The next result can be deduced from Theorem 8 in the same way that Corollary 5 in [10] was deduced from Theorem 4 of that paper. Corollary 14.…”

Section: Universal Laurent Seriesmentioning

confidence: 75%

“…The proofs of these results are directly analogous to the corresponding results for f 2 U(D; 0) (see Theorem 4 and Corollary 5 in [10]), except that, in the case of the corollary, Theorem 7 is used as a substitute for the fact that the Taylor series about 0 of a function in H(D) converges locally uniformly in D. They also rely on the observation that the boundary Harnack principle is valid in Jordan domains, by Carathéodory's theorem. Now let be an exterior Dini domain and 2 , choose 1 2 @ at maximum distance r from and let 1 = 1 + t( 1 )= j 1 j for some t > 0.…”

Section: Proof Of Theoremmentioning

confidence: 75%

“…Here we will adapt and combine arguments from [10] and [13]. Let , , 1 , 2 , r, E 1 , E 2 and be as in the statement of the theorem, and let E = E 1 [ E 2 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Gardiner

Manolaki

2014

Constr Approx

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Abstract. A holomorphic function on a planar domain is said to possess a universal Taylor series about a point of if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in Cn that have connected complement. In the case where is simply connected such functions are known to be unbounded, and to form a collection that is independent of the choice of . This paper uses tools from potential theory to show that, even for domains of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of . This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.

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“…The next result can be deduced from Theorem 8 in the same way that Corollary 5 in [10] was deduced from Theorem 4 of that paper. Corollary 14.…”

Section: Universal Laurent Seriesmentioning

confidence: 75%

Section: Proof Of Theoremmentioning

confidence: 75%

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Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Gardiner

Manolaki

2014

Constr Approx

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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%

“…The above corollaries are inspired by results recently established for universal Taylor series in [9,10]. In the version of Corollary 6 for universal Taylor series, an even stronger conclusion was subsequently established in [11], but its proof does not readily transfer to Dirichlet series.…”

Section: Applications To Universal Seriesmentioning

confidence: 93%

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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Taylor Series, Universality and Potential Theory

Gardiner

2018

Fields Institute Communications

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Universal Taylor series, conformal mappings and boundary behaviour

Cited by 14 publications

References 18 publications

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Boundary behaviour of Dirichlet series with applications to universal series

Taylor Series, Universality and Potential Theory

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