Comonotone approximation of periodic functions

Abstract

Suppose that a continuous 2π-periodic function f on the real axis ℝ changes its monotonicity at different ordered fixed points y _i ∈ [− π, π), i = 1, …, 2s, s ∈ ℕ. In other words, there is a set Y:= {y _i } _i∈ℤ of points y _i = y _i+2s + 2π on ℝ such that, on [y _i , y _i−1], f is nondecreasing if i is odd and nonincreasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that

$$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$

where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₂(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.