scholarly journals The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xia Shen ◽  
Hong Yan Xu
Keyword(s):  
Complex Plane ◽  
Infinite Order ◽  
Entire Functions ◽  
Point Of View ◽  
Type Function ◽  
Stieltjes Transforms

The main aim of this paper is to establish some theorems concerning the error E n F , β , the Sun’s type function U r , and M u σ , F of entire functions defined by Laplace-Stieltjes transforms with infinite order converge in the whole complex plane. Our results exhibit the growth of Laplace-Stieltjes transforms from the point of view of approximation.

scholarly journals The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Wen Ju Tang ◽  
Jian Chen ◽  
Hong Yan Xu
Keyword(s):  
Finite Order ◽  
Complex Plane ◽  
Infinite Order ◽  
Entire Functions ◽  
Growth Order ◽  
Stieltjes Transform ◽  
Proximate Order ◽  
Growth Of Entire Functions ◽  
Stieltjes Transforms

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

The Usual Behaviour of Rational Approximations

1983 ◽  
Vol 26 (3) ◽  
pp. 317-323 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Complex Plane ◽  
Metric Space ◽  
Unit Disc ◽  
Entire Functions ◽  
Complete Metric Space ◽  
Point Of View ◽  
Rational Approximations ◽  
Speed Of Convergence ◽  
Entire Plane ◽  
Categorical Point

AbstractQuestions concerning the convergence of Padé and best rational approximations are considered from a categorical point of view in the complete metric space of entire functions. The set of functions for which a subsequence of the mth row of the Padé table converges uniformly on compact subsets of the complex plane is shown to be residual.The speed of convergence of best uniform rational approximations and Padé approximations on the unit disc is compared. It is shown that, in a categorical sense, it is expected that subsequences of these approximants will converge at the same rate.Likewise, it is expected that the poles of certain sequences of best uniform rational approximations wil be dense in the entire plane.

scholarly journals Growth and Approximation of Laplace-Stieltjes Transform with p,q-Proximate Order Converges on the Whole Plane

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yong Qin Cui ◽  
Hong Yan Xu
Keyword(s):  
Complex Plane ◽  
Entire Functions ◽  
Equivalence Theorem ◽  
Stieltjes Transform ◽  
Proximate Order ◽  
Growth Of Entire Functions ◽  
Stieltjes Transforms

One purpose of this paper is to study the growth of entire functions defined by Laplace-Stieltjes transform converges on the whole complex plane, by introducing the concept of p,q-proximate order, and one equivalence theorem of the p,q-proximate order of Laplace-Stieltjes transforms is obtained. Besides, the second purpose of this paper is to investigate the approximation of entire functions defined by Laplace-Stieltjes transforms with p,q-proximate order, and some results about the p,q-proximate order, the error, and the coefficients of Laplace-Stieltjes transforms are obtained, which are generalization and improvement of the previous theorems given by Luo and Kong, Singhal, and Srivastava.

The phase problem

1976 ◽  
Vol 350 (1661) ◽  
pp. 191-212 ◽  
Keyword(s):  
Complex Plane ◽  
Hilbert Transform ◽  
Entire Functions ◽  
Experimental Methods ◽  
Point Of View ◽  
Phase Problem ◽  
Theoretical Interest ◽  
Phase Determination ◽  
Unique Character

The paper discusses the use of the theory of entire functions for solving the phase problem. In all practical cases only three forms of logarithmic Hilbert transform could possibly be required. The paper defines them and analyses their applicability. A generating form is also put forward for cases of possible theoretical interest. The uniqueness of the phase obtained from a logarithmic Hilbert transform is investigated and the difficulties due to the presence of zeros in the complex plane are discussed. Methods are put forward for both the removal of the zeros and, when this is not possible, for locating them in order to include their effect. The paper analyses known experimental methods for phase determination from the point of view of the theory presented and highlights their unique character.

On entire functions of infinite order

1963 ◽  
Vol 14 (1) ◽  
pp. 323-327 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Infinite Order ◽  
Entire Functions

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

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