APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

APPROXIMATION OF FUNCTIONS BY GAUSS-WEIERSTRASS INTEGRALS

2021 ◽  
Vol 2 ◽  
pp. 112-118
Author(s):  
Olga Shvai ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Partial Differential ◽  
Linear Positive Operator ◽  
Zygmund Class ◽  
Boundary Properties

When considering various schemes and algorithms for game problems of dynamics, researchers often have to deal with solutions of partial differential equations. A special place among the latter is occupied by the so-called equations of elliptic type (according to the corresponding classification), with the help of which natural and social processes can be described most fully and qualitatively. Moreover, the mathematical apparatus of partial differential equations of elliptic type makes it possible to get into the environment of deterministic phenomena and thus makes it possible to foresee their future. This fact undoubtedly increases the significance of the above type of equations among others in the sense of their application to mathematical modeling. At the same time, one of the most important concepts in applied mathematics is the concept of the modulus of continuity. The term "modulus of continuity" and its definition were introduced by Henri Lebesgue at the beginning of the last century in order to study various properties of continuous functions. Using the concept of the modulus of continuity and its properties, it is possible to investigate the belonging of the object under study to a certain class of functions: Hölder, Lipschitz, Zygmund, etc. This undoubtedly makes it possible to approximate functions of various kinds of operators most effectively. In this paper, using the example of the Gauss-Weierstrass integral as a solution to the corresponding differential equation of elliptic type, we study its rate of convergence in terms of the modulus of continuity of the second order to the function by which it was actually constructed. Namely, the boundary properties of the Gauss-Weierstrass integral were studied as a linear positive operator that realizes its best approximation on functions from the Zygmund class. The results obtained in this article can further be used to solve many problems in applied mathematics.

On a quantitative theory of limits: Estimating the speed of convergence

2020 ◽  
Vol 23 (4) ◽  
pp. 1013-1024
Author(s):  
Renato Spigler
Keyword(s):  
Modulus Of Continuity ◽  
Fractional Derivatives ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Speed Of Convergence ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Quantitative Theory ◽  
Hölder Continuous Functions ◽  
Definition Of

AbstractThe classical “ε-δ” definition of limits is of little use to quantitative purposes, as is needed, for instance, for computational and applied mathematics. Things change whenever a realistic and computable estimate of the function δ(ε) is available. This may be the case for Lipschitz continuous and Hölder continuous functions, or more generally for functions admitting of a modulus of continuity. This, provided that estimates for first derivatives, fractional derivatives, or the modulus of continuity might be obtained. Some examples are given.

ON APPROXIMATION OF FUNCTIONS FROM ZYGMUND CLASSES BY BIHARMONIC POISSON INTEGRALS

2021 ◽  
Vol 4 ◽  
Author(s):  
Bogdan Borsuk ◽  
◽  
Alexander Khanin ◽  
Keyword(s):  
Fourier Series ◽  
Applied Mathematics ◽  
Linear Operators ◽  
Systematic Research ◽  
Elliptic Type ◽  
Approximation Of Functions ◽  
Linear Positive Operators ◽  
Poisson Integrals ◽  
Direct Use ◽  
Poisson Type

The paper is devoted to a behavior investigation of the upper bound of deviation of functions from Zygmund classes from their biharmonic Poisson integrals. Systematic research in this direction was conducted by a number of Ukrainian as well as foreign scientists. But most of the known results relate to an estimation of deviations of functions from different classes from operators that were constructed based on triangular l-methods of the Fourier series summation (Fejer, Valle Poussin, Riesz, Rogozinsky, Steklov, Favard, etc.). Concerning the results relating to linear methods of the Fourier series summation, given by a set of functions of natural argument (Abel-Poisson, Gauss-Weierstrass, biharmonic and threeharmonic Poisson integrals), in this direction the progress was less notable. This may be due to the fact that the above-mentioned linear methods the Fourier series summation are solutions of corresponding integral and differential equations of elliptic type. And, therefore, they require more time-consuming calculations in order to obtain some estimates, that are suitable for a direct use for applied purposes. At the same time, in the present paper we investigate approximative characteristics of linear positive Poisson-type operators on Zygmund classes of functions. According to the well-known results by P.P. Korovkin, these positive linear operators realize the best asymptotic approximation of functions from Zygmund classes. Thus, the estimate obtained in this paper for the deviation of functions from Zygmund classes from their biharmonic Poisson integrals (the least studied and most valuable among all linear positive operators) is relevant from the viewpoint of applied mathematics.

Interspike interval dependency from arterial chemoreceptors

1985 ◽  
Vol 59 (5) ◽  
pp. 1566-1570 ◽  
Author(s):  
D. F. Donnelly ◽  
W. F. Nolan ◽  
E. J. Smith ◽  
R. E. Dutton
Keyword(s):  
Carotid Body ◽  
Interspike Interval ◽  
Correlation Coefficients ◽  
Feedback System ◽  
Serial Dependence ◽  
Control Series ◽  
First Order ◽  
System Operating ◽  
Poisson Type

The carotid body impulse generator has been previously characterized as a Poisson-type random process. We examined the validity of this characterization by analyzing sinus nerve spike trains for interspike interval dependency. Fifteen single chemoreceptive afferents were recorded in vivo under hypoxic-hypercapnic conditions, and approximately 1,000 consecutive interspike intervals for each fiber were timed and analyzed for serial dependence. The same set of intervals placed in shuffled order served as a control series without serial dependence. The original spike interval trains showed significantly negative first-order serial correlation coefficients and less variability in joint interval distributions than did the shuffled interval trains. These results suggest that the chemoreceptor afferent train is not random and may reflect a negative feedback system operating within the carotid body that limits variation about a mean frequency.

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

Elasticity Solution for a Radially Nonhomogeneous Hollow Circular Cylinder

1999 ◽  
Vol 66 (3) ◽  
pp. 598-606 ◽  
Author(s):  
Xiangzhou Zhang ◽  
Norio Hasebe
Keyword(s):  
Circular Cylinder ◽  
Continuous Functions ◽  
Variable Coefficients ◽  
Circular Cylinders ◽  
Explicit Solutions ◽  
Elasticity Solution ◽  
First Order ◽  
Hollow Circular Cylinder ◽  
Homogeneous Elasticity ◽  
Ordinary Differential Equation Systems

An exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio. In the solution, the cylinder is first approximated by a piecewise homogeneous one, of the same overall dimension and composed of perfectly bonded constituent homogeneous hollow circular cylinders. For each of the constituent cylinders, the solution can be obtained from the theory of homogeneous elasticity in terms of several constants. In the limit case when the number of the constituent cylinders becomes unboundedly large and their thickness tends to infinitesimally small, the piecewise homogeneous hollow circular cylinder reverts to the original nonhomogeneous one, and the constants contained in the solutions for the constituent cylinders turn into continuous functions. These functions, governed by some systems of first-order ordinary differential equations with variable coefficients, stand for the exact elasticity solution of the nonhomogeneous cylinder. Rigorous and explicit solutions are worked out for the ordinary differential equation systems, and used to generate a number of numerical results. It is indicated in the discussion that the developed method can also be applied to hollow circular cylinders with arbitrary, continuous radial nonhomogeneity.

scholarly journals On the integrability problem for systems of partial differential equations in one unknown function, I

2019 ◽  
Vol 11 (4) ◽  
pp. 35-71 ◽  
Author(s):  
Antonio Kumpera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Contact Structures ◽  
Partial Differential ◽  
Practical Applications ◽  
First Order ◽  
Integrability Problem ◽  
Integration Problem ◽  
In The Beginning

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

scholarly journals On multidimensional Jackson's theorem

2021 ◽  
pp. 30
Author(s):  
S.A. Pichugov
Keyword(s):  
Modulus Of Continuity ◽  
Periodic Functions ◽  
Linear Polynomial ◽  
Multiple Variables ◽  
Polynomial Methods ◽  
Methods Of Approximation

We have found the best linear polynomial methods of approximation of continuous periodic functions of multiple variables in uniform metric with concave modulus of continuity.

Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

2016 ◽  
Vol 60 (3) ◽  
pp. 615-633 ◽  
Author(s):  
Sonja Currie ◽  
Thomas T. Roth ◽  
Bruce A. Watson
Keyword(s):  
Periodic Potential ◽  
Adjoint System ◽  
Complex Potentials ◽  
Order System ◽  
Compact Sets ◽  
Periodic Functions ◽  
First Order ◽  
Adjoint Systems ◽  
Pauli Matrices

AbstractA self-adjoint first-order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π/2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ2Q(z)σ2. Here, Δ denotes the discriminant of the system and σ0, σ2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ0 + qσ2, for some real-valued π-periodic functions r and q integrable on compact sets.

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