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Harmonic extensions and the Böttcher-Silbermann conjecture

P. Gorkin, D. Zheng (1998)

Studia Mathematica

We present counterexamples to a conjecture of Böttcher and Silbermann on the asymptotic multiplicity of the Poisson kernel of the space L ( D ) and discuss conditions under which the Poisson kernel is asymptotically multiplicative.

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying s u p z D | f ( z ) | C ϱ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f.

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