Almost Coconvex Approximation of Continuous Periodic Functions

If a 2 𝜋 -periodic function f continuous on the real axis changes its convexity at 2s, s ∈ ℕ, inflection points y_i : − 𝜋 ≤ y_2s< y_2s−1< ... < y₁< 𝜋 and, for all other i ∈ ℤ, y_i are periodically defined, then, for any natural n ≥ Ny_i, we can find a trigonometric polynomial P_n of order c_n such that P_n has the same convexity as f everywhere except, possibly, small neighborhoods of the points y_i : (y_i− 𝜋/n, y_i + 𝜋/n) and, moreover,

$$ \left\Vert f-{P}_n\right\Vert \le c(s){\omega}_4\left(f,\uppi /n\right), $$

where Ny_i is a constant that depends only on min_i=1,...,2s{y_i− y_i+1}, c and c(s) are constants thatdepend only on s, 𝜔₄(f, ·) is the fourth modulus of smoothness of the function f, and ‖⋅‖is the max-norm.