Sets of determination and kernels of certain associated operators

2010 ◽  
Vol 53 (2) ◽  
pp. 503-510
Author(s):  
Arne Stray
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Closed Subset ◽  
Bounded Function ◽  
Harmonic Extension ◽  
Linear Map ◽  
Associated Operators

AbstractLet m be a measure supported on a relatively closed subset X of the unit disc. If f is a bounded function on the unit circle, let fm denote the restriction to X of the harmonic extension of f to the unit disc. We characterize those m such that the pre-adjoint of the linear map f → fm has a non-trivial kernel.

scholarly journals A Boundary Theorem for Tsuji Functions

1967 ◽  
Vol 29 ◽  
pp. 197-200 ◽  
Author(s):  
E. F. Collingwood
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Spherical Derivative

Let D denote the unit disc | z | <1, C the unit circle | z | = 1 and Cr the circle | z| = r. Corresponding to any function w(z) meromorphic in D we denote by w*(z) the spherical derivative

scholarly journals A note on the phase retrieval of holomorphic functions

2020 ◽  
pp. 1-8
Author(s):  
Rolando Perez
Keyword(s):  
Unit Circle ◽  
Connected Domain ◽  
Unit Disc ◽  
Phase Retrieval ◽  
Holomorphic Functions ◽  
Nevanlinna Class

Abstract We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

scholarly journals Eigenvalue Normalized Recurrent Neural Networks for Short Term Memory

2020 ◽  
Vol 34 (04) ◽  
pp. 4115-4122
Author(s):  
Kyle Helfrich ◽  
Qiang Ye
Keyword(s):  
Neural Networks ◽  
Unit Circle ◽  
Recurrent Neural Networks ◽  
Unit Disc ◽  
Gradient Descent ◽  
Short Term Memory ◽  
Short Term ◽  
Term Memory ◽  
Gradient Descent Algorithm

Several variants of recurrent neural networks (RNNs) with orthogonal or unitary recurrent matrices have recently been developed to mitigate the vanishing/exploding gradient problem and to model long-term dependencies of sequences. However, with the eigenvalues of the recurrent matrix on the unit circle, the recurrent state retains all input information which may unnecessarily consume model capacity. In this paper, we address this issue by proposing an architecture that expands upon an orthogonal/unitary RNN with a state that is generated by a recurrent matrix with eigenvalues in the unit disc. Any input to this state dissipates in time and is replaced with new inputs, simulating short-term memory. A gradient descent algorithm is derived for learning such a recurrent matrix. The resulting method, called the Eigenvalue Normalized RNN (ENRNN), is shown to be highly competitive in several experiments.

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 Numerical ranges of conjugations and antilinear operators on a Banach space

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2715-2720
Author(s):  
Muneo Chō ◽  
Injo Hur ◽  
Ji Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Circle ◽  
Banach Spaces ◽  
Unit Disc ◽  
Numerical Range ◽  
Space Setting ◽  
Path Connectedness ◽  
The Given

In this paper, we prove that the numerical range of a conjugation on Banach spaces, using the connected property, is either the unit circle or the unit disc depending the dimension of the given Banach space. When a Banach space is reflexive, we have the same result for the numerical range of a conjugation by applying path-connectedness which is applicable to the Hilbert space setting. In addition, we show that the numerical ranges of antilinear operators on Banach spaces are contained in annuli.

scholarly journals On minimal thinness, reduced functions and Green potentials

1993 ◽  
Vol 36 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Matts Essén
Keyword(s):  
Unit Circle ◽  
Recent Work ◽  
Unit Disc ◽  
Minimum Principle ◽  
Higher Dimensions ◽  
Connected Subset ◽  
Whitney Decomposition ◽  
Minimal Thinness ◽  
Image Position ◽  
Green Potentials

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we considerwhere Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.

scholarly journals Hp multipliers and inequalities of Hardy and Littlewood

1969 ◽  
Vol 10 (1-2) ◽  
pp. 23-32 ◽  
Author(s):  
G. I. Gaudry
Keyword(s):  
Unit Circle ◽  
Hardy Spaces ◽  
Unit Disc ◽  
Closed Ideal ◽  
Poisson Formula

Consider the classical Hardy spaces Hp(T) (1 ≦ p ≦ ∞) on the unit circle T. We shall ignore completely the fact that the elements of Hp(T) can be extended via the Poisson formula to certain types of functions analytic inside the unit disc. For our purposes, Hp(T) is the closed ideal in Lp(T) consisting of those functions f ∈ Lp(T) for which (n) = 0 (n= –1, –2,…).

Interpolation by Linear Sums of Harmonic Measures

1975 ◽  
Vol 18 (2) ◽  
pp. 249-253
Author(s):  
Marvin Ortel
Keyword(s):  
Unit Circle ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Boundary Values ◽  
Harmonic Measures ◽  
Image Position ◽  
Interior Points

Let α be an open arc on the unit circleand for z=reiθ, 0 ≤ r < 1, let(1)The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

scholarly journals The Existence and Structure of Rotational Systems in the Circle

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Jayakumar Ramanathan
Keyword(s):  
Dynamical System ◽  
Unit Circle ◽  
Closed Subset ◽  
Structure Theorem ◽  
Universal Cover ◽  
Invariant Sets ◽  
Continuous Transformation ◽  
Standard Orientation ◽  
Rotational Systems ◽  
Explicit Mention

By a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property that map f has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, F:T→T. Because our main result makes no explicit mention of a global transformation on T, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation F:T→T with finite preimages. In particular, there are no explicit conditions on the degree of F. We then give a development of known results in the case where Fθ=d·θmod⁡1 for an integer d>1. The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.

scholarly journals Extensions of Jensen's inequality on the unit circle

1986 ◽  
Vol 33 (2) ◽  
pp. 189-194
Author(s):  
Takahiko Nakazi
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Unit Disc ◽  
Jensen’S Inequality ◽  
Jensen's Inequality

The purpose of this note is to give extensions of an inequality of Jensen for analytic functions in an unit disc. We investigate functions which satisfy equalities in the inequalities.

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