scholarly journals Ideals of bounded holomorphic functions on simple n-sheeted discs

1991 ◽  
Vol 123 ◽  
pp. 171-201
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Banach Algebra ◽  
Positive Constant ◽  
Holomorphic Functions ◽  
Strong Sense ◽  
Supremum Norm ◽  
Corona Problem ◽  
Bounded Holomorphic ◽  
Zero Points ◽  
The Ideal

As usual we denote by H∞(K) the Banach algebra of bounded holomorphic functions on a Riemann surface R equipped with the supremum norm ‖·‖ Consider the ideal I(f1 … fm) of H∞(R) generated by functions f1 …fm in H∞(R). If a function g in H∞(R) belongs to I(f1 … fm or equivalently, if there exist m functions h1 …, hm in H∞(R) withon R, then common zero points of f1, ... fm are also zero points of g in the following strong sense:on R for a positive constant δ > 0. The generalized corona problem asks whether the converse is valid or not. In the case g ≡ 1 on R the problem is referred to simply as the corona problem.

A Note on Divisibility in H∞(X)

1984 ◽  
Vol 36 (3) ◽  
pp. 458-469 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Riemann Surface ◽  
Holomorphic Functions ◽  
Bounded Holomorphic

Let X be a Riemann surface, and H∞(X) the ring of bounded holomorphic functions in X. We offer here a question on divisibility in H∞(X), and then give in Section 2 a condition in which the answer is yes (Corollary 2 to Lemma 1). In Section 3 we use part 2 to prove a theorem on the separation of points by H∞(X). In Section 4 we study X/H∞(X).If f is meromorphic in X and z ∈ X, then by o(f, z) we mean the order of f at z. (We agree that o(f, z) = ∞ if f ≡ 0.) Let h be memomorphic in X; then h might be said to be of bounded type if h = f/g where f,g ∈ H∞(X), g ≠ 0.

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

2021 ◽  
pp. 1-29
Author(s):  
ALEXANDER BRUDNYI
Keyword(s):  
Disjoint Union ◽  
A Priori ◽  
Holomorphic Functions ◽  
Stable Rank ◽  
Open Unit ◽  
Uniform Estimates ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Similar Approximation ◽  
Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals A Riemann on the ideal boundary of a Riemann surface

1956 ◽  
Vol 32 (6) ◽  
pp. 409-411 ◽  
Author(s):  
Shin'ichi Mori ◽  
Minoru Ota
Keyword(s):  
Riemann Surface ◽  
Ideal Boundary ◽  
The Ideal

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

scholarly journals Dual Algebras and A-Measures

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Marek Kosiek ◽  
Krzysztof Rudol
Keyword(s):  
Analytic Functions ◽  
Complete Solution ◽  
Holomorphic Functions ◽  
Classical Case ◽  
Function Algebra ◽  
Domain Of Holomorphy ◽  
Strict Pseudoconvexity ◽  
Weak Star ◽  
Spectrum Of A Function ◽  
Bounded Holomorphic

Weak-star closures of Gleason parts in the spectrum of a function algebra are studied. These closures relate to the bidual algebra and turn out both closed and open subsets of a compact hyperstonean space. Moreover, weak-star closures of the corresponding bands of measures are reducing. Among the applications we have a complete solution of an abstract version of the problem, whether the set of nonnegative A-measures (called also Henkin measures) is closed with respect to the absolute continuity. When applied to the classical case of analytic functions on a domain of holomorphyΩ⊂Cn, our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity ofΩ. We also investigate the relation between the algebra of bounded holomorphic functions onΩand its abstract counterpart—thew* closure of a function algebraAin the dual of the band of measures generated by one of Gleason parts of the spectrum ofA.

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