Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

Abstract

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C ˇ;1 by Poisson integrals in the metric of the space C: Statement of the Problem and Some Historical InformationLet C be the space of 2 -periodic continuous functions with the norm kf k C D max t jf .t/j; let L 1 be the space of 2 -periodic, measurable, essentially bounded functions with the norm kf k 1 D ess sup t jf .t/j; and let L 1 be the space of 2 -periodic summable functions with the norm kf k L 1 D kf k 1 D Z jf .t/jdt:In 1983, Stepanets proposed a new approach to the classification of periodic functions. This approach is based on the notion of . ;ˇ/-derivative (see, e.g., [1][2][3][4]). The classes L ˇo f functions f 2 L 1 are introduced as follows: Let a sequence D .k/ and parameterˇbe such that the series 1 X kD1

Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals

“…where β r k are defined by the formula (14). From the formulas (17) and (30) we obtain the equality (13). Theorem 1 is proven.…”

“…Later, approximation properties of the Poisson integrals were investigated on other functional classes. In particular, we note the works [1]- [7], [13], [15]- [17].…”

“…Proof. According to the equality (16), it is sufficient to prove Theorem 2 only for the case of the uniform metric. In Timan's article [12] it was shown that…”

Approximation of (ψ, β)-differentiable functions of low smoothness by biharmonic poisson integrals

“…Following [17], we can prove that, for functions 2 M 0 ; both integrals on the right-hand side of (55) are of order O.1/ for ı ! 1; i.e., they are uniformly bounded with respect to ı: Thus, it follows from (53) and (55) that…”

Approximation of functions from the class $ C_{\beta, \infty }^\psi $ by Poisson integrals in the uniform metric

“…the Kolmogorov-Nikol'skii problem was solved in [7]. The aim of the present paper is to find asymptotic equalities for upper bounds of deviations of Poisson integrals on the classes C ˇ;1 forˇ2 R in the cases where 2 M C and 2 M 1 ; i.e., for functions .t / that decrease to zero as t !…”