On interpolation by analytic maps in infinite dimensions

1978 ◽  
Vol 83 (2) ◽  
pp. 243-252 ◽  
Author(s):  
J. Globevnik
Keyword(s):  
Unit Ball ◽  
Complex Banach Space ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Infinite Dimensions ◽  
Open Unit Ball ◽  
Infinite Dimensional ◽  
Analytic Maps ◽  
Peak Interpolation ◽  
Vector Valued Functions

AbstractLet A be the complex Banach algebra of all bounded continuous complex-valued functions on the closed unit ball of a complex Banach space X, analytic on the open unit ball, with sup norm. For a class of spaces X which contains all infinite dimensional complex reflexive spaces we prove the existence of non-compact peak interpolation sets for A. We prove some related interpolation theorems for vector-valued functions and present some applications to the ranges of analytic maps between Banach spaces. We also show that in general peak interpolation sets for A do not exist.

Boundaries for polydisc algebras in infinite dimensions

1979 ◽  
Vol 85 (2) ◽  
pp. 291-303 ◽  
Author(s):  
J. Globevnik
Keyword(s):  
Unit Ball ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Infinite Dimensions ◽  
Open Unit Ball ◽  
Image Position ◽  
Bounded Continuous Functions ◽  
Uniformly Continuous

AbstractLet AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) iffor every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

scholarly journals Universal functions in several complex variables

1979 ◽  
Vol 28 (2) ◽  
pp. 189-196 ◽  
Author(s):  
P. S. Chee
Keyword(s):  
Unit Ball ◽  
Universal Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Inner Function ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Blaschke Products ◽  
Universal Functions ◽  
Open Unit Ball

AbstractIt is proved that there exists a universal good inner function in the open unit polydisc Un, that is its non Euclidean translates are dense in the closed unit ball of H∞ (Un) and that there exists a universal function in the open unit ball Bn of Cn. These generalize Heins' result on universal Blaschke products.1980 Mathematics subject classification (Amer. Math. Soc.): primary 32 A 10.

Holomorphically embedded discs with rapidly growing area

1999 ◽  
Vol 129 (2) ◽  
pp. 343-349
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Open Unit ◽  
Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

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)

scholarly journals Complex extreme measurable selections

1995 ◽  
Vol 58 (2) ◽  
pp. 222-231
Author(s):  
Douglas Mupasiri
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Closed Unit Ball ◽  
Measurable Selections ◽  
Necessary And Sufficient

AbstractWe give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < 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.

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

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