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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.

Lacunary series in Q K spaces

Hasi Wulan, Kehe Zhu (2007)

Studia Mathematica

Under mild conditions on the weight function K we characterize lacunary series in the so-called K spaces.

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