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Associated weights and spaces of holomorphic functions

Klaus Bierstedt, José Bonet, Jari Taskinen (1998)

Studia Mathematica

When treating spaces of holomorphic functions with growth conditions, one is led to introduce associated weights. In our main theorem we characterize, in terms of the sequence of associated weights, several properties of weighted (LB)-spaces of holomorphic functions on an open subset G ⊂ ℂ N which play an important role in the projective description problem. A number of relevant examples are provided, and a “new projective description problem” is posed. The proof of our main result can also serve to characterize...

Asymptotic values and the growth of analytic functions in spiral domains.

James E. Brennan, Alexander L. Volberg (1993)

Publicacions Matemàtiques

In this note we present a simple proof of a theorem of Hornblower which characterizes those functions analytic in the open unit disk having asymptotic values at a dense set in the boundary. Our method is based on a kind of ∂-mollification and may be of use in other problems as well.

Attractors with vanishing rotation number

Rafael Ortega, Francisco Ruiz del Portal (2011)

Journal of the European Mathematical Society

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carath´eodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

Pietro Poggi-Corradini (2003)

Revista Matemática Iberoamericana

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a...

Baker domains for Newton’s method

Walter Bergweiler, David Drasin, James K. Langley (2007)

Annales de l’institut Fourier

For an entire function f let N ( z ) = z - f ( z ) / f ′ ( z ) be the Newton function associated to f . Each zero ξ of f is an attractive fixed point of N and is contained in an invariant component of the Fatou set of the meromorphic function N in which the iterates of N converge to ξ . If f has an asymptotic representation f ( z ) ∼ exp ( - z n ) , n ∈ ℕ , in a sector | arg z | < ε , then there exists an invariant component of the Fatou set where the iterates of N tend to infinity. Such a component is called an invariant Baker domain.A question in the opposite direction...

Basic relations valid for the Bernstein spaces B ² σ and their extensions to larger function spaces via a unified distance concept

P. L. Butzer, R. L. Stens, G. Schmeisser (2014)

Banach Center Publications

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces B σ p are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from B σ p . The difficult situation of derivative-free error estimates is also covered.

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