On the Zeros of Functions with Derivatives in H1 and H∞

1970 ◽  
Vol 22 (2) ◽  
pp. 342-347 ◽  
Author(s):  
James Wells
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Open Unit ◽  
Hardy Class ◽  
Open Unit Disc ◽  
Zeros Of Functions ◽  
Limit Points ◽  
Image Position ◽  
Derivatives Of

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying(A)and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.THEOREM 1. If(1)(2)and(3)then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

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.

6.—Rationally Convex Hulls and Potential Theory.

1976 ◽  
Vol 74 ◽  
pp. 81-89
Author(s):  
Richard F. Basener
Keyword(s):  
Potential Theory ◽  
Compact Subset ◽  
Unit Disc ◽  
Rational Functions ◽  
Continuous Functions ◽  
Uniform Algebra ◽  
Convex Hulls ◽  
Open Unit ◽  
Open Unit Disc ◽  
Image Position

SynopsisLet S be a compact subset of the open unit disc in C. Associate to S the setLet R(X) be the uniform algebra on X generated by the rational functions which are holomorphic near X. It is shown that the spectrum of R(X) is determined in a simple wayby the potential-theoretic properties of S. In particular, the spectrum of R(X) is X if and only if the functions harmonic near S are uniformly dense in the continuous functions on S. Similar results can be obtained for other subsets of C2 constructed from compact subsets of C.

scholarly journals On the Asymptotic Behavior of Functions Harmonic in a Disc

1966 ◽  
Vol 28 ◽  
pp. 187-191
Author(s):  
J. E. Mcmillan
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Continuous Curve ◽  
Open Unit Disc ◽  
Boundary Path

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

scholarly journals Pointwise bounded approximation by polynomials

1992 ◽  
Vol 112 (1) ◽  
pp. 147-155 ◽  
Author(s):  
Anthony G. O'Farrell ◽  
Fernando Perez-Gonzalez
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Bounded Analytic Functions ◽  
Unbounded Component ◽  
Open Set ◽  
Approximation By Polynomials ◽  
Polynomial Hull ◽  
Image Position

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

Ideals in Rings of Analytic Functions with Smooth Boundary Values

1970 ◽  
Vol 22 (6) ◽  
pp. 1266-1283 ◽  
Author(s):  
B. A. Taylor ◽  
D. L. Williams
Keyword(s):  
Banach Algebra ◽  
Unit Disc ◽  
Smooth Boundary ◽  
Topological Algebra ◽  
Boundary Function ◽  
Open Unit ◽  
Boundary Values ◽  
Open Unit Disc ◽  
Algebra Of Functions ◽  
Derivatives Of

LetAdenote the Banach algebra of functions analytic in the open unit discDand continuous in. Iffand its firstmderivatives belong toA,then the boundary functionf(eiθ)belongs toCm(∂D). The spaceAmof all such functions is a Banach algebra with the topology induced byCm(∂D).If all the derivatives of/ belong toA,then the boundary function belongs toC∞(∂D), and the spaceA∞all such functions is a topological algebra with the topology induced byC∞(∂D). In this paper we determine the structure of the closed ideals ofA∞(Theorem 5.3).Beurling and Rudin (see e.g. [7, pp. 82-89;10]) have characterized the closed ideals ofA, and their solution suggests a possible structure for the closed ideals ofA∞.

scholarly journals Convergence criteria for bounded sequences

1972 ◽  
Vol 18 (2) ◽  
pp. 99-103 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Complex Numbers ◽  
Convergence Criteria ◽  
Open Unit Disc ◽  
Primary Object ◽  
Image Position ◽  
Bounded Sequences

Let {Kn} be a sequence of complex numbers, letand letLet D be the open unit disc {z: |z| <1}, let be its closure and let .The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).

scholarly journals The Closed Ideals in an Algebra of Analytic Functions

1957 ◽  
Vol 9 ◽  
pp. 426-434 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Complex Functions ◽  
Continuous Complex ◽  
Image Position

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).

scholarly journals Vanishing l1-sums of the Poisson kernel, and sums with positive coefficients

1989 ◽  
Vol 32 (3) ◽  
pp. 431-447 ◽  
Author(s):  
F. F. Bonsall ◽  
D. Walsh
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disc ◽  
Image Position ◽  
Positive Coefficients

For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − z̄ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn withby which we mean that the series converges to zero in the norm of L1(∂D).

scholarly journals Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

1987 ◽  
Vol 30 (3) ◽  
pp. 471-477 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Harmonic Function ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Open Unit ◽  
Countable Subset ◽  
Open Unit Disc ◽  
Image Position ◽  
Bounded Harmonic Functions ◽  
Bounded Harmonic Function

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

scholarly journals On a conjecture of Z. Jianzhong

1992 ◽  
Vol 45 (1) ◽  
pp. 163-170
Author(s):  
Yasuo Matsugu
Keyword(s):  
Convex Function ◽  
Unit Ball ◽  
Unit Circle ◽  
Orlicz Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Almost All

Let ϕ be a nonnegative, nondecreasing and nonconstant function defined on [0, ∞) such that Φ(t) = ϕ(et) is a convex function on (-∞, ∞). The Hardy-Orlicz space H (ϕ) is defined to be the class of all those functions f holomorphic in the open unit disc of the complex plane C satisfying The subclass H(ϕ)+ of H(ϕ) is defined to be the class of all those functions f ∈ H(ϕ) satisfying for almost all points eit of the unit circle. In 1990, Z. Jianzhong conjectured that H(ϕ)+ = H(ψ)+ if and only if H(ϕ) = H(ψ). In the present paper we prove that it is true not only on the unit disc of C but also on the unit ball of Cn.

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