On Density Conditions for Interpolation in the Ball

2003 ◽  
Vol 46 (4) ◽  
pp. 559-574 ◽  
Author(s):  
Nicolas Marco ◽  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Sufficient Condition ◽  
Density Condition ◽  
Interpolating Sequences ◽  
Class A ◽  
Spaces Of Holomorphic Functions

AbstractIn this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of Cn, n > 1. We first give density conditions for a sequence to be interpolating for the class A−∞ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for n = 1 coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

scholarly journals A–∞-interpolation in the ball

1998 ◽  
Vol 41 (2) ◽  
pp. 359-367 ◽  
Author(s):  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Derivatives Of

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

On Classes for Hyperbolic Riemann Surfaces

2016 ◽  
Vol 59 (01) ◽  
pp. 13-29
Author(s):  
Rauno Aulaskari ◽  
Huaihui Chen
Keyword(s):  
Unit Ball ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Inclusion Relations ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractThe Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.

scholarly journals Holomorphic Spaces in the Unit Ball of

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Augusto Guadalupe Miss Paredes ◽  
Lino Feliciano Reséndis Ocampo ◽  
Luis Manuel Tovar Sánchez
Keyword(s):  
Unit Ball ◽  
Holomorphic Functions ◽  
Vector Spaces ◽  
Spaces Of Holomorphic Functions ◽  
Harmonic Majorants

We introduce the and vector spaces of holomorphic functions defined in the unit ball of , generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000).

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