Approximative properties of the three-harmonic Poisson integrals on the classes $$ {W}_{\beta}^r{H}^{\alpha } $$

Approximation of Classes $$ {C}_{\beta, \infty}^{\psi } $$ by Three-Harmonic Poisson Integrals in Uniform Metric (Low Smoothness)

“…It is known (see, e.g., [7,11]) that the values of approximation by the Poisson and biharmonic Poisson integrals cannot tend to zero at δ → ∞ faster than 1 δ and 1 δ 2 , respectively. At the same time, as follows from works [17][18][19], using the approximation by three-harmonic integrals, one can obtain the approximation rate 1 δ 3 , δ → ∞. Therefore, our purpose is to study the asymptotic behavior of the quantities E(C ψ β,∞ ; P 3,δ ) C at δ → ∞.…”

“…The first results related to the study of the approximative properties of three-harmonic Poisson integrals were obtained in work [6]. Later, the research in this direction was continued in works [17][18][19]. In particular, the Kolmogorov-Nikolskii problem for the three-harmonic Poisson integral on the classes W r β,∞ , r > 0, β ∈ R, was solved in [17], and on the classes [18,19].…”

“…Later, the research in this direction was continued in works [17][18][19]. In particular, the Kolmogorov-Nikolskii problem for the three-harmonic Poisson integral on the classes W r β,∞ , r > 0, β ∈ R, was solved in [17], and on the classes [18,19]. As concerning the study of the approximative properties of three-harmonic Poisson integrals on the classes of (ψ, β)-differentiable functions, this issue has not been studied enough.…”