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Bases communes holomorphes: nouvelle extension du théorème de Whittaker

Nguyen Thanh Van, Patrice Lassere (1993)

Annales Polonici Mathematici

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 ∂ D α ⊂ D E , D α 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 O ( D α ) .

Boundary behavior and Cesàro means of universal Taylor series.

Frédéric Bayart (2006)

Revista Matemática Complutense

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.

Boundary functions in L 2 H ( 𝔹 n )

Piotr Kot (2007)

Czechoslovak Mathematical Journal

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function u which is lower semi-continuous on ∂ 𝔹 n we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ 𝕆 ( 𝔹 n ) such that u ( z ) = ∫ | λ | < 1 f ( λ z ) 2 d 𝔏 2 ( λ ) .

Boundary functions on a bounded balanced domain

Piotr Kot (2009)

Czechoslovak Mathematical Journal

We solve the following Dirichlet problem on the bounded balanced domain Ω with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂ Ω with u ( z ) = u ( λ z ) for | λ | = 1 , z ∈ ∂ Ω we construct a holomorphic function f ∈ 𝕆 ( Ω ) such that u ( z ) = ∫ 𝔻 z | f | p d 𝔏 𝔻 z 2 for z ∈ ∂ Ω , where 𝔻 = { λ ∈ ℂ | λ | < 1 } .

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