scholarly journals Approximation of the periodical functions of high smoothness by the rightangled Fourier sums

2013 ◽  
Vol 5 (1) ◽  
pp. 102-109
Author(s):  
O.O. Novikov ◽  
O.G. Rovenska
Keyword(s):  
Upper Bounds ◽  
Functions Of Two Variables ◽  
High Smoothness ◽  
Fourier Sums ◽  
The Right

We obtain asymptotic equalities for upper bounds of the deviations of the right-angled Fourier sums taken over classes of periodical functions of two variables of high smoothness. These equalities in corresponding cases guarantee the solvability of the Kolmogorov–Nikol’skii problem for the right-angled Fourier sums on the specified classes of functions.

scholarly journals Some (p,q)-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 683 ◽  
Author(s):  
Humaira Kalsoom ◽  
Muhammad Amer ◽  
Moin-ud-Din Junjua ◽  
Sabir Hussain ◽  
Gullnaz Shahzadi
Keyword(s):  
Convex Functions ◽  
Integral Type ◽  
Partial Derivatives ◽  
Functions Of Two Variables ◽  
Right Hand ◽  
The Absolute ◽  
The Right ◽  
Quasi Convexity

In this paper, we present the preliminaries of ( p , q ) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the ( p , q ) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the ( p , q ) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the ( p , q ) -calculus of the functions of two variables.

scholarly journals Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

2021 ◽  
pp. 12-17
Author(s):  
Oleg Lytvyn ◽  
Oleg Lytvyn ◽  
Oleksandra Lytvyn
Keyword(s):  
Differentiable Function ◽  
Fourier Coefficients ◽  
Gibbs Phenomenon ◽  
Discontinuous Function ◽  
Projection Data ◽  
Approximation Accuracy ◽  
Discontinuous Functions ◽  
Functions Of Two Variables ◽  
Fourier Sums ◽  
The Difference

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

scholarly journals New continuity estimates of geometric sums

2002 ◽  
Vol 15 (3) ◽  
pp. 219-233 ◽  
Author(s):  
Evgueni Gordienko ◽  
Juan Ruiz de Chávez
Keyword(s):  
Stochastic Models ◽  
Random Number ◽  
Random Variables ◽  
Upper Bounds ◽  
Uniform Distance ◽  
Right Hand ◽  
The Stability ◽  
The Right ◽  
Geometric Sums ◽  
Independent And Identically Distributed

The paper deals with sums of a random number of independent and identically distributed random variables. More specifically, we compare two such sums, which differ from each other in the distributions of their summands. New upper bounds (inequalities) for the uniform distance between distributions of sums are established. The right-hand sides of these inequalities are expressed in terms of Zolotarev's and the uniform distances between the distributions of summands. Such a feature makes it possible to consider these inequalities as continuity estimates and to apply them to the study of the stability (continuity) of various applied stochastic models involving geometric sums and their generalizations.

Bounds for Arakelov invariants of modular curves

2011 ◽  
Author(s):  
Bas Edixhoven ◽  
Robin de Jong
Keyword(s):  
Upper Bounds ◽  
Green Functions ◽  
Modular Curves ◽  
Right Hand ◽  
The Right

This chapter gives bounds for all quantities on the right-hand side in the inequality in Theorems 9.1.1 and 9.2.5, in the context of the modular curves X₁(5l) with l > 5 prime, using the upper bounds for Green functions from Chapter 10. The final estimates are given in the last section of the chapter.

scholarly journals Moments and oscillations of exponential sums related to cusp forms

2016 ◽  
Vol 162 (3) ◽  
pp. 479-506 ◽  
Author(s):  
ESA V. VESALAINEN
Keyword(s):  
Exponential Sums ◽  
Upper Bounds ◽  
Cusp Forms ◽  
Average Order ◽  
Small Denominators ◽  
Moment Estimates ◽  
Holomorphic Cusp Forms ◽  
Order Of Magnitude ◽  
The Mean ◽  
The Right

AbstractWe consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twistse(nh/k) with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, thek-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form.Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length 3π/2+ϵ and so cannot wind around the origin.

scholarly journals Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

2018 ◽  
Vol 32 ◽  
pp. 92-103
Author(s):  
Oleg Novikov ◽  
Olga Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Integral Representations ◽  
Linear Operators ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
Differentiable Functions ◽  
Fourier Sums ◽  
Linear Means

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

scholarly journals APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

2020 ◽  
Vol 8 (2) ◽  
pp. 114-121
Author(s):  
O. Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Periodic Functions ◽  
Arithmetic Means ◽  
Poisson Integrals ◽  
Fourier Sums ◽  
Uniformly Convergent

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

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