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.

Robust multiscale analytic sampling approximation to periodic function and fast algorithm

2017 ◽  
Vol 15 (01) ◽  
pp. 1750006
Author(s):  
Youfa Li ◽  
Jing Shang ◽  
Gengrong Zhang ◽  
Pei Dang
Keyword(s):  
Hardy Space ◽  
Periodic Function ◽  
Complex Plane ◽  
Time Domain ◽  
Fast Algorithm ◽  
Unit Disc ◽  
Numerical Experiments ◽  
Approximation Error ◽  
Hankel Matrix ◽  
Sample Error

By applying the multiscale method to the Möbius transformation function, we construct the multiscale analytic sampling approximation (MASA) to any function in the Hardy space [Formula: see text]. The approximation error is estimated, and it is proved that the MASA is robust to sample error. We prove that the MASA can be expressed by a Hankel matrix, making use of which, a fast algorithm is established to compute the MASA. Since what we acquire in practice may well be the samples on time domain instead of the analytic ones on the unit disc of the complex plane, we establish a fast algorithm for acquiring analytic samples. Numerical experiments are carried out to demonstrate the efficiency of the MASA.

scholarly journals Best approximation of holomorphic functions from hardy space in terms of Taylor coefficients

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1417-1424
Author(s):  
F.G. Abdullayev ◽  
G.A. Abdullayev ◽  
V.V. Savchuk
Keyword(s):  
Hardy Space ◽  
Polynomial Approximation ◽  
Best Approximation ◽  
Holomorphic Functions ◽  
Best Polynomial Approximation ◽  
Taylor Coefficients

We describe the set of holomorphic functions from the Hardy space Hq, 1 ? q ? ?, for which the best polynomial approximation En(f)q is equal to |f (n)(0)|=n!.

scholarly journals Bounded holomorphic projections for exponentially decreasing weights

2008 ◽  
Vol 6 (1) ◽  
pp. 59-70 ◽  
Author(s):  
Wolfgang Lusky ◽  
Jari Taskinen
Keyword(s):  
Large Class ◽  
Complex Plane ◽  
Unit Disc ◽  
Bergman Projections ◽  
Bounded Holomorphic

We construct generalized Bergman projections on a large class of weightedL∞–spaces. The examples include exponentially decreasing weights on the unit disc and complex plane.

ON MULTIPLIERS BETWEEN SOME SPACES OF HOLOMORPHIC FUNCTIONS

2017 ◽  
Vol 96 (1) ◽  
pp. 146-153
Author(s):  
BARTOSZ STANIÓW
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Toeplitz Matrices ◽  
Holomorphic Functions ◽  
Sequence Spaces ◽  
Spaces Of Holomorphic Functions

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence $\ell ^{p}$ spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on $\ell ^{p}$’, Sibirsk. Mat. Zh.7 (1966), 146–158].

scholarly journals The extreme points of a class of functions with positive real part

1991 ◽  
Vol 44 (2) ◽  
pp. 253-261
Author(s):  
N. Samaris
Keyword(s):  
Extreme Point ◽  
Unit Disc ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Greatest Common Divisor ◽  
Positive Real ◽  
Taylor Coefficients ◽  
Class Of Functions

Let P1 be the class of holomorphic functions on the unit disc U = {z: |z| < 1} for which f(0) = 1 and Re f > 0. Let also Pn be the corresponding class on the unit disc Un. The inequality |ak| ≤ 2 is known for the Taylor coefficients in the class P1. In this paper, it is generalised for the class Pn. If ρ = (ρ1, ρ2, …, ρn), with ρ1, ρ2, …, ρn nonegative integers whose greatest common divisor is equal to 1, we describe the form of the functions f ∈ Pn under the restriction |aρ| = 2. Under the same restriction, we give conditions for a function to be an extreme point of the class Pn.

scholarly journals Every Separable Complex Fréchet Space with a Continuous Norm is Isomorphic to a Space of Holomorphic Functions

2020 ◽  
pp. 1-5
Author(s):  
José Bonet
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Fréchet Space ◽  
Dense Subspace ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Infinite Dimensional ◽  
Spaces Of Holomorphic Functions

Abstract Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

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 On an F-Algebra of Holomorphic Functions

1988 ◽  
Vol 40 (3) ◽  
pp. 718-741 ◽  
Author(s):  
Hong Oh Kim
Keyword(s):  
Complex Plane ◽  
Positive Constant ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Maximal Theorem ◽  
Image Position ◽  
Limiting Case

The complex maximal theorem of Hardy and Little-wood states:(Mp). For 0 < p < ∞, there exists a positive constant Cp such that if f is holomorphic in the unit disc U of the complex plane thenwhereThe corresponding statement to the limiting case p = 0 can be stated as follows:(M0) There exists a positive constant C0 such that if f is holomorphic in Uwhere log+t = max(log t, 0).The statement (M0) is false as the following example shows.

Sign in / Sign up

Export Citation Format

Share Document