Multiple modules of continuity and the best approximations of periodic functions in metric spaces
It is proved that, under the condition M_Ψ ( 12) < 1, where M_Ψ is a stretching function Ψ in the space L_Ψ , the Jackson inequalities _n _fL_Ψ, f ot = const E_n-1(f)_Ψ ω_k (f, πn )_Ψ < ∞, are true; here, E_n-1(f)_Ψ is the best approximation of f by trigonometric polynomials of degree at most n - 1 and ω_k (f, πn )_Ψ is the modulus of continuity of f of order k, k N. We study necessary and sufficient conditions for the function f under which the following relation is true: E_n-1(f)_Ψ ω_k (f, πn )_Ψ., number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Pichugov, S. A., year=2018, month=May, pages=699–707