Multi-universal series

Mouze, Augustin

doi:10.1007/s00013-020-01467-y

Cited by 2 publications

(7 citation statements)

References 10 publications

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“…One may now ask whether this property is also shared by sequences T K or T K ρ , for some ρ = (r n ) n ⊂ [0, 1) tending to 1 and K ⊂ T different from T. In fact, we will constructively prove that this is not the case: given K 0 ⊂ T different from T and ρ = (r n ) n , there exist two functions f 1 , f 2 ∈ H(D), none of them Abel universal for T K0 , such that the set {T K ρ,n ( f i ) ∶ n ∈ N, i = 1, 2} is dense in C(K) for any compact set K ⊂ T different from T. As a consequence, this will also show that the families T K and T K ρ do not satisfy a Bourdon-Feldman-type property. We should mention that an analogue of those results was obtained by the second author for Fekete universal series, a realanalytic variant of universal Taylor series [35]. However, the case of universal Taylor series-i.e., for the sequences S K , K ⊂ C/D with connected complement-is still open.…”

Section: Introductionsupporting

confidence: 58%

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Abel universal functions

Charpentier¹,

Mouze²

2022

Can. J. Math.-J. Can. Math.

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Given a sequence = ( ) ∈ [0, 1) tending to 1, we consider the set U (D, ) of Abel universal functions consisting of holomorphic functions in the open unit disc D such that for any compact set included in the unit circle T, different from T, the set { ↦ → ( •) | : ∈ N} is dense in the space C ( ) of continuous functions on . It is known that the set U (D, ) is residual in (D). We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C ( ) for any compact set ⊂ T different from T. Moreover we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally an Abel universal function can be viewed as a universal vector of the sequence of dilation operators : ↦ → ( •) acting on (D). Thus we study the dynamical properties of ( ) such as the multi-universality and the (common) frequent universality. All the proofs are constructive.In complex function theory, it is of great interest to distinguish and study classes of holomorphic functions with a regular boundary behaviour. Regular boundary behaviour at a point of T can mean, for instance, convergence, Cesàro summability or Abel summability of the Taylor expansion at this point. We recall that a function in (D) is Abel summable at ∈ T if the quantity ( ), ∈ [0, 1), has a finite limit as → 1. Note that in the latter case, the cluster set of along the radius { : ∈ [0, 1)} is reduced to a single value. Yet, it is now well understood that non-regularity is a generic behaviour. We say that a property is generic in a Baire space if the set of those ∈ which satisfy

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Section: Introductionsupporting

confidence: 58%

“…The concept of multi-universality was recently studied in the case of Fekete universal series [35]. In this section, we are going to investigate it for Abel universal functions.…”

Section: Abel Multi-universalitymentioning

confidence: 99%

Abel universal functions

Charpentier¹,

Mouze²

2022

Can. J. Math.-J. Can. Math.

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“…As a consequence, this will also show that the families T K and T K ̺ do not satisfy a Bourdon-Feldman type property. We should mention that an analogue of those results was obtained by the second author for Fekete universal series, a real-analytic variant of universal Taylor series [31]. However, the case of universal Taylor series -i.e.…”

Section: Introductionsupporting

confidence: 57%

“…The concept of multi-universality was recently studied in the case of Fekete universal series [31]. In this section, we are going to investigate it for Abel universal series.…”

Section: Abel Multi-universalitymentioning

confidence: 99%

See 1 more Smart Citation

Abel universal series

Charpentier¹,

Mouze²

2022

Preprint

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Given a sequence ̺ = (r n ) n ∈ [0, 1) tending to 1, we consider the set U A (D, ̺) of Abel universal series consisting of holomorphic functions f in the open unit disc D such that for any compact set K included in the unit circle T, different from T, the set {z →. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C(K) for any compact set K ⊂ T different from T. Moreover we prove that the class of Abel universal series is not invariant under the action of the differentiation operator. Finally an Abel universal series can be viewed as a universal vector of the sequence of dilation operators T n : f → f (r n •) acting on H(D). Thus we study the dynamical properties of (T n ) n such as the multi-universality and the (common) frequent universality. All the proofs are constructive.

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Multi-universal series

Cited by 2 publications

References 10 publications

Abel universal functions

Abel universal functions

Abel universal series

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