scholarly journals Rate of Convergence of Hermite-Fejér Interpolation on the Unit Circle

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. Díaz ◽  
E. Martínez-Brey
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Complex Number ◽  
Analytic Functions ◽  
Order Of Convergence ◽  
Supremum Norm ◽  
Laurent Polynomials ◽  
Nodal System ◽  
Norm Of The Error

The paper deals with the order of convergence of the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system, thenroots of a complex number with modulus one. The supremum norm of the error of interpolation is obtained for analytic functions as well as the corresponding asymptotic constants.

scholarly journals Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 498
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Alberto Castejón ◽  
José Manuel García-Amor
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Lagrange Interpolation ◽  
Roots Of Unity ◽  
Smooth Functions ◽  
Numerical Examples ◽  
Practical Applications ◽  
Bounded Interval ◽  
Interesting Approach ◽  
Nodal System

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

scholarly journals Hermite Interpolation on the Unit Circle Considering up to the Second Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Jaime Díaz
Keyword(s):  
Fourier Transform ◽  
Unit Circle ◽  
Hermite Interpolation ◽  
Interpolation Polynomial ◽  
Complex Numbers ◽  
Second Derivative ◽  
Laurent Polynomials ◽  
Natural Basis ◽  
Barycentric Formula ◽  
Nodal System

The paper is devoted to study the Hermite interpolation problem on the unit circle. The interpolation conditions prefix the values of the polynomial and its first two derivatives at the nodal points and the nodal system is constituted by complex numbers equally spaced on the unit circle. We solve the problem in the space of Laurent polynomials by giving two different expressions for the interpolation polynomial. The first one is given in terms of the natural basis of Laurent polynomials and the remarkable fact is that the coefficients can be computed in an easy and efficient way by means of the Fast Fourier Transform (FFT). The second expression is a barycentric formula, which is very suitable for computational purposes.

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

scholarly journals The arc-length variation of analytic capacity and a conformal geometry

1992 ◽  
Vol 125 ◽  
pp. 151-216
Author(s):  
Takafumi Murai
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Conformal Geometry ◽  
Supremum Norm ◽  
Length Variation ◽  
Analytic Capacity ◽  
Arc Length ◽  
Bounded Analytic Functions ◽  
Extended Complex

For a domain Ω in the extended complex plane C ∪{∞}, H∞(Ω) denotes the Banach space of bounded analytic functions in Ω with supremum norm ∥ · ∥H∞ For ζ ∈ Ω, we putwhere f′(∞) = lim,z→∞z{f (∞) = f(z)} if ζ = ∞.

scholarly journals Biduals of weighted banach spaces of analytic functions

1993 ◽  
Vol 54 (1) ◽  
pp. 70-79 ◽  
Author(s):  
K. D. Bierstedt ◽  
W. H. Summers
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Open Subset ◽  
Analytic Functions ◽  
Holomorphic Functions ◽  
Supremum Norm ◽  
Closed Unit Ball ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions ◽  
Continuous Weight

AbstractFor a positive continuous weight function ν on an open subset G of CN, let Hv(G) and Hv0(G) denote the Banach spaces (under the weighted supremum norm) of all holomorphic functions f on G such that ν f is bounded and ν f vanishes at infinity, respectively. We address the biduality problem as to when Hν(G) is naturally isometrically isomorphic to Hν0(G)**, and show in particular that this is the case whenever the closed unit ball in Hν0(G) in compact-open dense in the closed unit ball of Hν(G).

scholarly journals Non-extreme-Points Approach to Extreme Points of Integral Families of Analytic Functions

10.53733/87 ◽  
2021 ◽  
Vol 51 ◽  
pp. 39-48
Author(s):  
Keiko Dow
Keyword(s):  
Unit Circle ◽  
Extreme Point ◽  
Analytic Functions ◽  
Compact Convex ◽  
Extreme Points ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Kernel Functions ◽  
Closed Convex Hull ◽  
Special Cases

Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying “all” the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has beenidentified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points.

scholarly journals Approximation by complex modified Szász-Mirakjan-Stancu operators in compact disks

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1007-1019 ◽  
Author(s):  
Nursel Çetin
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Analytic Functions ◽  
Exact Order ◽  
Order Of Approximation ◽  
Stancu Operators ◽  
Exact Order Of Approximation ◽  
Compact Disks

In this paper, we establish some theorems on approximation and Voronovskaja type results for complex modified Sz?sz-Mirakjan-Stancu operators attached to analytic functions having exponential growth on compact disks. Also, we estimate the rate of convergence and the exact order of approximation.

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