On the derivatives of bounded analytic functions

»ke Samuelsson

doi:10.1007/bf02591129

Asymptotic Behavior of Fractional Derivatives of Bounded Analytic Functions

Zurnal matematiceskoj fiziki analiza geometrii ◽

10.15407/mag13.02.107 ◽

2017 ◽

Vol 13 (2) ◽

pp. 107-118

Author(s):

I. Chyzhykov ◽

◽

Yu. Kosaniak ◽

Keyword(s):

Asymptotic Behavior ◽

Analytic Functions ◽

Fractional Derivatives ◽

Bounded Analytic Functions ◽

Derivatives Of

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The space of totally bounded analytic functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028303 ◽

1987 ◽

Vol 30 (2) ◽

pp. 229-246

Author(s):

Alan L. Horwitz ◽

Lee A. Rubel

Keyword(s):

Analytic Function ◽

Analytic Functions ◽

Bounded Analytic Functions ◽

Sharp Bounds ◽

Totally Bounded ◽

Bounded Set ◽

Inverse Interpolation ◽

Derivatives Of ◽

Uniformly Bounded

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

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About the Optimal Recovery of Derivatives of Analytic Functions from Their Values at Points That Form a Regular Polygon

Mathematical Physics and Computer Simulation ◽

10.15688/mpcm.jvolsu.2019.4.2 ◽

2019 ◽

pp. 30-38

Author(s):

Mikhail Ovchintsev

Keyword(s):

Approximation Method ◽

Blaschke Product ◽

Analytic Functions ◽

Best Approximation ◽

Extremal Function ◽

Regular Polygon ◽

Optimal Recovery ◽

Bounded Analytic Functions ◽

The Best Approximation ◽

Derivatives Of

In this paper, the author solves the problem of optimal recovery of derivatives of bounded analytic functions defined at the zero of the unit circle. Recovery is performed based on information about the values of these functions at points z1, ... , zn , that form a regular polygon. The article consists of an introduction and two sections. The introduction talks about the necessary concepts and results from the works of Osipenko K.Yu. and Khavinson S.Ya., that form the basis for the solution of the problem. In the first section, the author proves some properties of the Blaschke product with zeros at the points z1, ... , zn. After this, the error of the best approximation method of the derivatives f(N)(0), 1 ≤ N ≤ n − 1, by the values f(z1), ... , f(zn) is calculated. In the same section he gives the corresponding extremal function. In the second section, the uniqueness of the linear best approximation method is established, and then its coefficients are calculated. At the end of the article, the formulas found for calculating of the coefficients are substantially simplified.

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A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-2-7 ◽

2020 ◽

Vol 17 (2) ◽

pp. 256-277

Author(s):

Ol'ga Veselovska ◽

Veronika Dostoina

Keyword(s):

Complex Plane ◽

Complex Variable ◽

Analytic Functions ◽

Combinatorial Identities ◽

Independent Interest ◽

Closed Curves ◽

In Series ◽

Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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