Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions

Abstract

We consider a 2π-periodic function f continuous on $\mathbb{R}$ and changing its sign at 2s points y _i ∈ [−π, π). For this function, we prove the existence of a trigonometric polynomial T _n of degree ≤n that changes its sign at the same points y _i and is such that the deviation | f(x) − T _n(x) | satisfies the second Jackson inequality.