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On convergence sets of divergent power series

Buma L. Fridman, Daowei Ma, Tejinder S. Neelon (2012)

Annales Polonici Mathematici

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family y = φ s ( t , x ) = s b ₁ ( x ) t + b ₂ ( x ) t ² + ⋯ of analytic curves in ℂ × ℂⁿ passing through the origin, C o n v φ ( f ) of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series f ( φ s ( t , x ) , t , x ) converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that E = C o n v φ ( f ) if and only if...

On highly nonintegrable functions and homogeneous polynomials

P. Wojtaszczyk (1997)

Annales Polonici Mathematici

We construct a sequence of homogeneous polynomials on the unit ball d in ℂ d which are big at each point of the unit sphere . As an application we construct a holomorphic function on d which is not integrable with any power on the intersection of d with any complex subspace.

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: | ∑ n = 0 ∞ a ₙ z ⁿ | < 1 , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: | ∑ n = 0 k a ₙ z ⁿ | < 1 , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are considered....

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