Stechkin-Type Estimate for Nearly Copositive Approximations of Periodic Functions

Abstract

Assume that a continuous 2𝜋 -periodic function f defined on the real axis changes its sign at 2s, s ∈ ℕ, points yi : −𝜋 ≤ y2s N(k, yi), where N(k, yi) is a constant that depends only on k ∈ ℕ and mini=1,...,2s{yi − yi+1}, we construct a trigonometric polynomial Pn of order ≤ n, which has the same sign as f everywhere, except (possibly) small neighborhoods of the points yi : (yi − π/n, yi + π/n), Pn(yi) = 0, i ∈ ℤ, and in addition, ‖f − Pn‖ ≤ c(k, s)ωk(f, π/n), where c(k, s) is a constant that depends only on k and s, 𝜔k(f, ·) is the k th modulus of smoothness of f, and ∥·∥ is the max-norm.