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We introduce the concept of asymptotic stability for a set of complex functions analytic around the origin, implicitly contained in an earlier paper of the first mentioned author (“Finite group actions and asymptotic expansion of ", Combinatorica 17 (1997), 523 – 554). As a consequence of our main result we find that the collection of entire functions with the set of all real polynomials satisfying Hayman’s condition is asymptotically stable. This answers a question raised in loc. cit.
Résumé. Soient D un ouvert de ℂ et E un compact de D. Moyennant une hypothèse assez faible sur D et ℂ̅ E on montre que si α ∈ ]0,1[ vérifie , étant l’ouvert de niveau z ∈ D : ω(E,D,z) < α, alors toute base commune de O(E) et O(D) est une base de .
We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its Cesàro means are universal.
We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function such that
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For and a positive lower semi-continuous function on with for , we construct a holomorphic function such that for , where .
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