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 The Radon Inversion Problem for Holomorphic Functions in the Unit Disc

2021 ◽  
Author(s):  
P. Kot ◽  
P. Pierzchała
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Positive Function ◽  
Inversion Problem ◽  
Technical Improvement ◽  
Taylor Coefficients ◽  
Convergence And Divergence ◽  
Strictly Positive Function

AbstractThis paper deals with the so-called Radon inversion problem formulated in the following way: Given a $$p>0$$ p > 0 and a strictly positive function H continuous on the unit circle $${\partial {\mathbb {D}}}$$ ∂ D , find a function f holomorphic in the unit disc $${\mathbb {D}}$$ D such that $$\int _0^1|f(zt)|^pdt=H(z)$$ ∫ 0 1 | f ( z t ) | p d t = H ( z ) for $$z \in {\partial {\mathbb {D}}}$$ z ∈ ∂ D . We prove solvability of the problem under consideration. For $$p=2$$ p = 2 , a technical improvement of the main result related to convergence and divergence of certain series of Taylor coefficients is obtained.

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

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

Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions

1973 ◽  
Vol 25 (4) ◽  
pp. 812-819
Author(s):  
David C. Haddad
Keyword(s):  
Complex Number ◽  
Unit Disc ◽  
Boundary Behavior ◽  
Dense Subset ◽  
Holomorphic Functions ◽  
Asymptotic Values

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

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 Some remarks on angular ranges and sequences of ρ-points for holomorphic functions

1975 ◽  
Vol 58 ◽  
pp. 69-82
Author(s):  
H. Yoshida
Keyword(s):  
Unit Disc ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Spherical Derivative

In this paper, we will give examples of holomorphic functions in the unit disc having singular connections between the growth of maximum modulus and angular ranges (Theorem A) as well as singular connections between the growth of spherical derivative and sequences of ρ-points (Theorem B).

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.

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