Exact constants in Jackson inequalities for periodic differentiable functions in the space L _∞

Abstract

It is proved that, in the space L _∞[0, 2π], the following equalities hold for all k = 0, 1, 2, …, n ∈ ℕ, r = 1, 3, 5, …, µ≥ r:

where E _n−1(f) and E _n,µ (f) are the best approximations of f by, respectively, trigonometric polynomials of degree n − 1 and 2π-periodic splines of minimal deficiency of order µ with 2n equidistant nodes, ω(f ^(r), h) is the modulus of continuity of f ^(r), Ψ _r,2k+1 is the rth periodic integral of the special function Ψ _0,2k+1, which is odd and piecewise constant on the partition jπ/(2k + 1), j ∈ ℤ. For k = 0, this result was obtained earlier by Ligun.