Sharp Constant in Jackson’s Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions

Abstract

EN: Abstract—It is proved that, in the space C2π, for all k,n ∈ N, n > 1, the following inequalities hold:

1 − 1 2n k2 + 1 2 ≤ sup f∈C2π f=const en−1(f) ω2(f,π/(2nk)) ≤ k2 + 1 2 . where en−1(f) is the value of the best approximation of f by trigonometric polynomials and ω2(f,h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.