Overconvergent series of rational functions and universal Laurent series

Müller, Jürgen; Vlachou, Vagia; Yavrian, A.

doi:10.1007/s11854-008-0023-7

Cited by 12 publications

(6 citation statements)

References 8 publications

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“…In 2005, Bayart [44] established the dense-lineability in H(D) of the class of these functions f , which are known as universal Taylor series. Since 1996, many extensions of these results, for other domains and more restrictive classes of functions, have been performed (see the recent papers [30,49,53,132,228] and references therein). They can be put into a more general context.…”

Section: Hypercyclity and Dense-lineability An Extreme Case Of Lineamentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

195

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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Section: Hypercyclity and Dense-lineability An Extreme Case Of Lineamentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

195

149

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“…This notion was introduced by Costakis, Nestoridis and Papadoperakis [5], who showed that the collection U L ( ; 1 ; :::; k ) of functions with this property is a dense G subset of H( ). Subsequent work on universal Laurent series includes [6], [12], [15], [16] and [20].…”

Section: Universal Laurent Seriesmentioning

confidence: 99%

“…Next, let K A j nf j g be a compact set with connected complement, and let g 2 C(K) \ H(K ). By Theorem 5 of [16] we can choose a sequence (N k ) in N such that…”

Section: Universal Laurent Seriesmentioning

confidence: 99%

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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“…Sthn perÐptwsh aut , to er¸thma sqetik me thn Ôparxh kajolik¸n seir¸n Taylor eÐnai pio polÔploko, kurÐwc epeid to sÔnolo twn poluwnÔmwn den eÐnai puknì ston antÐstoiqo q¸ro twn olìmorfwn sunart sewn. Gia èna mh apl sunektikì tìpo Ω, tou opoÐou to sumpl rwma perièqei mÐa mh fragmènh sunist¸sa , isqÔei ìti U(Ω, ξ) = ∅ gia kje epilog kèntrou ξ ∈ Ω (blèpe [54], [51] kai [34]). Apì thn llh pleur, arketoÐ sug-grafeÐc apèdeixan thn Ôparxh kajolik¸n seir¸n Taylor se exeidikeumènouc mh apl sunektikoÔc tìpouc (blèpe [73], [50], [24], [12], [29], [61], [5]).…”

Section: H Apìdeixh Thcunclassified

“…Apì thn llh pleur, arketoÐ sug-grafeÐc apèdeixan thn Ôparxh kajolik¸n seir¸n Taylor se exeidikeumènouc mh apl sunektikoÔc tìpouc (blèpe [73], [50], [24], [12], [29], [61], [5]). Ja jèlame na epishmnoume, ìti sthn perÐptwsh aut h klsh U(Ω, ξ) exarttai genik apì to ξ (koÐta [50]), prgma pou den isqÔei ìtan to Ω eÐnai apl sunektikì (koÐta [54]).…”

Section: H Apìdeixh Thcunclassified

Καθολικές σειρές Taylor σε διπλά συνεκτικούς τόπους, καθολικές συναρτήσεις και υπερκυκλικότητα

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Overconvergent series of rational functions and universal Laurent series

Cited by 12 publications

References 8 publications

Linear subsets of nonlinear sets in topological vector spaces

Linear subsets of nonlinear sets in topological vector spaces

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Καθολικές σειρές Taylor σε διπλά συνεκτικούς τόπους, καθολικές συναρτήσεις και υπερκυκλικότητα

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