Generic Behavior of Classes of Taylor Series Outside the Unit Disk

Let E be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space H 2 for which the partial sums of its Taylor series approximate any continuous function on E. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in H 2 instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to Dirichlet-type spaces D α for α ∈ [0, 1].

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“…on sets that are not necessarily contained in the unit circle. Results in this direction for the Taylor case have been obtained in [11].…”

Section: Concluding Remarks and Further Directionsmentioning

confidence: 99%

Simultaneous zero-free approximation and universal optimal polynomial approximants

Bénéteau

Ivrii

Manolaki

et al. 2020

Journal of Approximation Theory

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“…The functions in U(D) were extensively studied from many points of view. We refer to [1,4,12,18,20,21,22,29,30,32] and the references therein for a non-exhaustive list of papers. In particular, some of these highlight the fact that functions in U(D) enjoy irregular boundary behaviour, for example along radii.…”

Section: Introductionmentioning

confidence: 99%

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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“…The functions in U (D) were extensively studied from many points of view. We refer to [3,6,15,21,23,24,25,33,34,36] and the references therein for a non-exhaustive list of papers. In particular, some of these highlight the fact that functions in U (D) enjoy irregular boundary behaviour, for example along radii.…”

mentioning

confidence: 99%

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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Generic Behavior of Classes of Taylor Series Outside the Unit Disk

Cited by 7 publications

References 19 publications

Simultaneous zero-free approximation and universal optimal polynomial approximants

Simultaneous zero-free approximation and universal optimal polynomial approximants

Abel universal series

Abel universal functions

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