scholarly journals Interpolation by holomorphic functions in the unit ball with polynomial growth

1997 ◽  
Vol 6 (2) ◽  
pp. 277-296 ◽  
Author(s):  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Holomorphic Functions

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.

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

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 ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

2011 ◽  
Vol 85 (2) ◽  
pp. 307-314 ◽  
Author(s):  
ZHANGJIAN HU
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Holomorphic Functions ◽  
Essential Norm ◽  
Normal Weight ◽  
Cesàro Operator ◽  
Radial Derivative ◽  
Image Position ◽  
Cesaro Operator

AbstractLet Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

scholarly journals The strict topology on spaces of bounded holomorphic functions

1994 ◽  
Vol 49 (2) ◽  
pp. 249-256 ◽  
Author(s):  
Juan Ferrera ◽  
Angeles Prieto
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Strict Topology ◽  
Open Unit Ball ◽  
Approximation By Polynomials ◽  
Bounded Holomorphic

We introduce in this paper the space of bounded holomorphic functions on the open unit ball of a Banach space endowed with the strict topology. Some good properties of this topology are obtained. As applications, we prove some results on approximation by polynomials and a description of the continuous homomorphisms.

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