A NOTE ON ZEROS OF BOUNDED HOLOMORPHIC FUNCTIONS IN WEAKLY PSEUDOCONVEX DOMAINS IN ℂ2

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 $H^\infty ({\mathbb {D}})$ .

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A note on a space Hp, a of holomorphic functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013447 ◽

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.

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Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-026-2 ◽

1962 ◽

Vol 14 ◽

pp. 334-348 ◽

Cited By ~ 16

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).

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