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).

Some Lemmas on Interpolating Blaschke Products and a Correction

1969 ◽  
Vol 21 ◽  
pp. 531-534 ◽  
Author(s):  
A. Kerr-Lawson
Keyword(s):  
Blaschke Product ◽  
Unit Disc ◽  
Inner Function ◽  
Open Unit ◽  
Inner Functions ◽  
Singular Measure ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Interpolating Blaschke Products

A Blaschke product on the unit disc,where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the conditionCis satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product andfor some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metricand its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

The Maximal Ideal Space of H∞ + C on the ball in Cn

1979 ◽  
Vol 31 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Gerard Mcdonald
Keyword(s):  
Maximal Ideal ◽  
Banach Algebras ◽  
Holomorphic Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Open Unit Ball ◽  
Closed Subalgebra ◽  
Image Position ◽  
Bounded Holomorphic

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limitsexist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

Integral Means and Zero Distributions of Blaschke Products

1972 ◽  
Vol 24 (5) ◽  
pp. 755-760 ◽  
Author(s):  
C. N. Linden
Keyword(s):  
Half Plane ◽  
Blaschke Product ◽  
Blaschke Products ◽  
Integral Means ◽  
Elementary Theorem ◽  
Image Position ◽  
Analogous Theorem

A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only ifIf 0 appears m times in {zn} thenis the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such thatas r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

On the Growth of Blaschke Products

1969 ◽  
Vol 21 ◽  
pp. 595-601 ◽  
Author(s):  
G. R. MacLane ◽  
L. A. Rubel
Keyword(s):  
Angular Distribution ◽  
Analytic Function ◽  
Blaschke Product ◽  
Bounded Function ◽  
Related Question ◽  
Complex Numbers ◽  
Private Communication ◽  
Blaschke Products ◽  
The Subject ◽  
Image Position

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product1Let2What are the sequences {zn} for which I(r) is a bounded function of r?

Almost-Bounded Holomorphic Functions with Prescribed Ambiguous Points

1964 ◽  
Vol 16 ◽  
pp. 231-240 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Jordan Arcs ◽  
Ambiguous Point

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that

Zero Tracts of Blaschke Products

1966 ◽  
Vol 18 ◽  
pp. 1072-1078 ◽  
Author(s):  
C. N. Linden ◽  
H. Somadasa
Keyword(s):  
Unit Disk ◽  
Blaschke Product ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disk ◽  
Blaschke Products ◽  
Image Position

Let ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.

Composition with a Nonhomogeneous Bounded Holomorphic Function on the Ball

1989 ◽  
Vol 41 (5) ◽  
pp. 870-881
Author(s):  
Jun Soo Choa ◽  
Hong Oh Kim
Keyword(s):  
Holomorphic Functions ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Homogeneous Polynomials ◽  
Open Unit Disc ◽  
Mean Oscillation ◽  
Bounded Holomorphic Function ◽  
Euclidean Unit Ball ◽  
Image Position ◽  
Bounded Holomorphic

For an integer n > 1, the letters U and Bn denote the open unit disc in C and the open euclidean unit ball in Cn, respectively. It is known that the homogeneous polynomialswhere bα is chosen so that , have the following pull-back property:If g ∈ ℬ(U) the Block space, then , the space of holomorphic functions on Bn of bounded mean oscillation, forand.

Tangential H∞-images of boundary curves

1988 ◽  
Vol 104 (1) ◽  
pp. 115-118 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Borel Measure ◽  
Holomorphic Functions ◽  
Continuous Family ◽  
Open Set ◽  
Finite Borel Measure ◽  
Image Position ◽  
Boundary Curves ◽  
Bounded Holomorphic ◽  
Fatou’S Theorem

1.Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixedn≥ 1. We define (Γ, μ) to be aFatou pair inΩ if(a) Γ is a continuous family of boundary curves γwin Ω, one ending at each w ∈ ∂Ω,(b) μ is a positive finite Borel measure on ∂Ω, and(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:(a) The map(w, t) → γw(t)is continuous, from ∂Ω × [0, 1) into Ω, andfor every w in the boundary ∂Ω of Ω.(c) For everyf ∈ H∞(Ω)(the class of all bounded holomorphic functions in Ω), the limitexists a.e. [μ].

Radial variation of analytic functions with non-tangential boundary limits almost everywhere

1990 ◽  
Vol 108 (2) ◽  
pp. 371-379 ◽  
Author(s):  
D. J. Hallenbeck ◽  
K. Samotij
Keyword(s):  
Analytic Function ◽  
Asymptotic Behaviour ◽  
Unit Disk ◽  
Analytic Functions ◽  
Radial Variation ◽  
The Third ◽  
Almost Everywhere ◽  
Image Position ◽  
Boundary Limits ◽  
Tangential Limits

The purpose of this paper is to investigate the asymptotic behaviour as r → 1− of the integralsand f is an analytic function on the unit disk Δ which has non-tangential limits at almost every point on ∂Δ. The paper is divided into three parts. In the first part we consider the case where λ ≠ 1/k, in the second the somewhat more delicate case when λ = 1/k and in the third part we concentrate on some problems related to the case λ = k = 1.

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