Jackson's Theorem Research Articles

A note of some approximation theorems of functions on the Laguerre hypergroup

This paper uses some basic notions and results on the Laguerre hypergroup K = [0, +?)xR to study some problems in the theory of approximation of functions in the space L2 ?(K). Analogues of the direct Jackson theorems of approximations for the modulus of smoothness (of arbitrary order) constructed by using the generalized translation operators on K are proved. The Nikolskii-Stechkin inequality is also obtained. In conclusion of this work, we show that the modulus of smoothness and the K-functionals constructed from the Sobolev-type space corresponding to the Laguerre operator L? are equivalent.

Jackson Theorems for the Quaternion Linear Canonical transform

Jackson Theorems for the Quaternion Linear Canonical transform

On multidimensional Jackson's theorem

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.

On approximation of functions by algebraic polynomials in $L^p_{\rho}$ metric

The version has been found, of improved Jackson's theorem analogue, concerning the approximation by algebraic polynomials on the interval in the integral metric for some classes of functions being integrable with the following weight: $\rho(x) = (1-x)^{\alpha} (1+x)^{\beta}$.

A note on [formula omitted]-functional, Modulus of smoothness, Jackson theorem and Bernstein–Nikolskii–Stechkin inequality on Damek–Ricci spaces

In this paper we study approximation theorems for L2-space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K-functional and modulus of smoothness for Damek–Ricci spaces.

Polynomial approximation in several variables

This paper is about the characterization of the rate of best polynomial approximation on a domain in Rd by multivariate polynomials of degree n via a suitable modulus of continuity or smoothness. We shall be concerned with the situations: approximation on convex polytopes and approximation on (not necessarily convex) smooth algebraic domains. We shall introduce appropriate moduli of smoothness and prove the corresponding Jackson theorem. The usual weak converses are also true.

Fourier–Jacobi harmonic analysis and some problems of approximation of functions on the half-axis in L2 metric: Jackson's type direct theorems

ABSTRACTWe use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions in weighted function spaces on the half-axis . We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. As a tool for approximation, we use functions with bounded spectrum.

On Jackson Type Inequalities for the Best Approximations of Periodic Functions

UDC 517.5 We develop a new approach to the proof of a generalized Jackson theorem providing an estimate for the best approximation by trigonometric polynomials with the help of the moduli of continuity of an arbitrary order. The proposed approach is based on approximation methods constructed by using Fejer and Jackson–Vallee - Poussin classical kernels. Bibliography :1 4titles.

Certain approximation problems for functions on the infinite-dimensional torus: Analogs of the Jackson theorem

Certain approximation problems for functions on the infinite-dimensional torus: Analogs of the Jackson theorem

Constants in Jackson Type Inequalities for the Best Approximation of Periodic Functions Such that Some of Their Fourier Coefficients Vanish

We consider the class of continuous 2π-periodic functions such that some of their Fourier coefficients vanish. For such functions we study constants in a generalized Jackson theorem providing an estimate for the best approximation by trigonometric polynomials with the help of the moduli of continuity of an arbitrary order.

Jackson networks with single-line nodes and limited sojourn or waiting times

We consider an exponential queueing network with single-line nodes that differs from a Jackson network only in that the time claims sojourn in the system (waiting time for the servicing) at the network's nodes is limited by an exponentially distributed random value. Claims serviced at nodes and claims that have not finished servicing (that exceeded maximal waiting time) move along the network according to different routing matrices. We prove a theorem that generalizes Jackson's theorem for open and closed networks.

Analysis of New Services Influence for Signaling Network Node

Before the introduction of new services in the telecommunication network, need to estimate the influence of the realization algorithm of service and possible the intensity of usage on the fitness for work of the signaling network nodes. Each service has own interaction between intelligent network node and different number of transactions. In the work gives the analytical model whish include service switching point/intelligent peripheral nodes (SSP/IP), service control point/service data point (SCP/SDP) and the nodes of signaling network (STP). For creating the analytical IN model was used Jackson's theorem. By using the proposed model, is cared out the calculation of the influence of services free phone FP and Account Calling Card ACC on signalling nodes of the given network. Ill. 7, bibl. 8 (in Lithuanian; summaries in Lithuanian, English, Russian).

On Constants in the Generalized Jackson Theorem

We consider the generalized Jackson theorem concerning estimates for the best approximation by trigonometric polynomials via the moduli of continuity of an arbitrary order. We improve the known estimates for moduli of continuity of the 4th and 6th order. Bibliography: 8 titles.

Jackson's Theorem in Qp Spaces in Polydiscs

Jackson's theorem is an important result in the theory of approximation of function. The purpose of this article is to introduce a new kind of Qp spaces in the unit polydiscs. Jackson's approximation theorem is established in Qp spaces.

Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier-Haar series

The paper deals with the space consisting of classes of real measurable functions on with finite integral . If , then the space can be made into a Banach space with the norm . The inequality , which is an analogue of the first Jackson theorem, is shown to hold for the finite Fourier-Haar series , provided that the variable exponent satisfies the condition . Here, is the modulus of continuity in defined in terms of Steklov functions. If the function lies in the Sobolev space with variable exponent , it is shown that . Methods for estimating the deviation for at a given point are also examined. The value of is calculated in the case when , where .Bibliography: 17 titles.

Fourier–Jacobi harmonic analysis and approximation of functions

We use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions by algebraic polynomials in weighted function spaces on . We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. The moduli of smoothness are shown to be equivalent to -functionals constructed from Sobolev-type spaces. We define Nikol'skii–Besov spaces for the Jacobi generalized translation and describe them in terms of best approximations. We also prove analogues of some inverse theorems of Stechkin.

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight \(u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}\), α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.

Approximation by algebraic polynomials in metric spaces $L_{\psi}$

In the space $L_{\psi}[-1;1]$ of non-periodic functions with metric $\rho(f,0)_{\psi} = \int\limits_{-1}^1 \psi(|f(x)|)dx$, where $\psi$ is a function of the type of modulus of continuity, we study Jackson inequality for modulus of continuity of $k$-th order in the case of approximation by algebraic polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

Inequalities of the type of the generalized Jackson theorem for the best approximations

Very simple methods for deriving upper estimates of the type of the generalized Jackson theorem for the best approximations of periodic functions are proposed, The results obtained are similar to the generalized Jackson theorem, providing estimates in terms of the moduli of continuity. Bibliography: 12 titles.

Approximation of functions in by trigonometric polynomials

We consider the Lebesgue space with variable exponent . It consists of measurable functions for which the integral exists. We establish an analogue of Jackson's first theorem in the case when the -periodic variable exponent satisfies the conditionUnder the additional assumption we also get an analogue of Jackson's second theorem. We establish an -analogue of Bernstein's estimate for the derivative of a trigonometric polynomial and use it to prove an inverse theorem for the analogues of the Lipschitz classes for . Thus we establish direct and inverse theorems of the theory of approximation by trigonometric polynomials in the classes . In the definition of the modulus of continuity of a function , we replace the ordinary shift by an averaged shift determined by Steklov's function .