scholarly journals A Localization Principle for a Class of Analytic Functions

1963 ◽  
Vol 23 ◽  
pp. 207-212
Author(s):  
D. A. Storvick
Keyword(s):  
Analytic Function ◽  
Circular Disk ◽  
Inverse Image ◽  
Limit Values ◽  
Localization Principle ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Simply Connected Domain ◽  
Almost Everywhere ◽  
Simply Connected

It has been shown by Kiyoshi Noshiro [8; p. 35] that a bounded analytic function w = f(z) in |z| < 1 having radial limit values of modulus one almost everywhere satisfies a localization principle of the following type. Let (c) be any circular disk: | w − α | < ρ lying inside |w| < 1 whose periphery may be tangent to the circumference |w| = 1. Denote by Δ any component of the inverse image of (c) under w = f(z) and by z = z(ξ) a function which maps |ξ| < 1 onto the simply connected domain Δ in a one-to-one conformal manner. Then, the functionis also a bounded analytic function in | ξ | < 1 with radial limits of modulus one almost everywhere.

scholarly journals On harmonic continuation

1963 ◽  
Vol 6 (1) ◽  
pp. 54-56
Author(s):  
M. S. P. Eastham
Keyword(s):  
Analytic Function ◽  
Finite Number ◽  
Connected Domain ◽  
Arc Length ◽  
Jordan Curves ◽  
Simply Connected Domain ◽  
Harmonic Continuation ◽  
Simply Connected

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.

scholarly journals Bounded analytic functions and the little Bloch space

1990 ◽  
Vol 13 (1) ◽  
pp. 193-198 ◽  
Author(s):  
Rohan Attele
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Analytic Functions ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Little Bloch Space

The radial limits of the weighted derivative of an bounded analytic function is considered.

scholarly journals A variational method for the construction of convergent iterative sequences

1986 ◽  
Vol 41 (1) ◽  
pp. 51-58 ◽  
Author(s):  
Zalman Rubinstein
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Analytic Function ◽  
Variational Method ◽  
Connected Domain ◽  
Initial Point ◽  
Proper Subset ◽  
Simply Connected Domain ◽  
Arbitrary Initial Point ◽  
Simply Connected

AbstractConvergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

On the Two Questions of Lohwater and Piranian

1996 ◽  
Vol 3 (5) ◽  
pp. 447-456
Author(s):  
M. Gvaradze
Keyword(s):  
Analytic Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Zero Measure ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Necessary And Sufficient

Abstract The problem we are dealing with consists in the following: find the necessary and sufficient conditions for the zero measure subset of the circumference at which points the bounded analytic function has no radial limits.

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 Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

scholarly journals On the Relation between a Cluster set Introduced by Constantinescu and Cornea, and the Fine Cluster set of Cartan, Brelot, and Naïm

1965 ◽  
Vol 8 (1) ◽  
pp. 59-71
Author(s):  
H. L. Jackson
Keyword(s):  
Analytic Function ◽  
Holomorphic Function ◽  
Limit Theorems ◽  
Function Theory ◽  
Angular Limit ◽  
Boundary Limit ◽  
Cluster Set ◽  
Almost Everywhere ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic

The field of boundary limit theorems in analytic function theory is usually considered to have begun about 1906, with the publication of Fatou's thesis [8]. In this remarkable memoir a theorem is proved, that now bears the author's name, which implies that any bounded holomorphic function defined on the unit disk possesses an angular limit almost everywhere (Lebesgue measure) on the frontier. Outstanding classical contributions to this field can be attributed to F. and M. Riesz, R. Nevanlinna, Lusin, Privaloff, Frostman, Plessner, and others.

Sign in / Sign up

Export Citation Format

Share Document