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.

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 A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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.

Some remarks on universal functions and Taylor series

2000 ◽  
Vol 128 (1) ◽  
pp. 157-175 ◽  
Author(s):  
G. COSTAKIS
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
Connected Domain ◽  
Constructive Proof ◽  
Universal Functions ◽  
Baire’S Theorem ◽  
Simply Connected Domain ◽  
Universal Taylor Series ◽  
Simply Connected

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

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