APPROXIMATION OF FUNCTIONS BY GAUSS-WEIERSTRASS INTEGRALS

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.

Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

The nonlocal boundary value problem for the parabolic differential equationv'(t)+A(t)v(t)=f(t) (0≤t≤T), v(0)=v(λ)+φ, 0<λ≤Tin an arbitrary Banach spaceEwith the dependent linear positive operatorA(t)is investigated. The well-posedness of this problem is established in Banach spacesC0β,γ(Eα-β)of allEα-β-valued continuous functionsφ(t)on[0,T]satisfying a Hölder condition with a weight(t+τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

Some applications of Shisha-Mond theorem

A result due to Shisha, O. and Mond, B., is recalled and some applications in the evaluation of approximation order by linear positive operator are presented.

Asymptotic expressions for the remainder term in the quadrature formula of Gauss-Jacobi type

In this paper we have considered error analysis for a quadrature formula which is obtained by integration of linear positive operator. The asymptotic expressions for remainder term of Gauss-Jacobi type quadrature formula are also given.

Théorème ergodique pour les opérateurs positifs à moyennes bornées sur les espaces Lp(1 < p < ∞)

The main result is a dominated ergodic theorem for a linear positive operator T on Lp(1 > p > ∞); the theorem holds if, and only if, T is Cesaro-bounded.

Minimal convergence on Lp spaces

Let (X, F, μ) be a probability measure space, p and β real numbers such that 1≤p<+∞ and 0<β<p. For any linear positive operator T satisfying T1, T*1 = 1 we prove the norm and pointwise convergence of the sequence We get then the pointwise and norm convergence in Lp, 0 < β ≥ 1 < p < 2, of the sequence sgn Sif for any positive linear operator on Lp(Ω, A, μ) (μ-σ-finite) verifying ∥(1 − α)I + αS∥p ≤ 1 for a real number 0 < α < 1. In the particular case α = 1, (S is a contraction), β = p−l, this result gives the pointwise and norm convergence of the sequences introduced by Beauzamy and Enflo in 1985 to the asymptotic center of the sequence .