Almost Coconvex Approximation of Continuous Periodic Functions

Abstract

If a 2 𝜋 -periodic function f continuous on the real axis changes its convexity at 2s, s ∈ ℕ, inflection points yi : − 𝜋 ≤ y2s< y2s−1< ... < y1< 𝜋 and, for all other i ∈ ℤ, yi are periodically defined, then, for any natural n ≥ Nyi, we can find a trigonometric polynomial Pn of order cn such that Pn has the same convexity as f everywhere except, possibly, small neighborhoods of the points yi : (yi− 𝜋/n, yi + 𝜋/n) and, moreover,$$ \left\Vert f-{P}_n\right\Vert \le c(s){\omega}_4\left(f,\uppi /n\right), $$ where Nyi is a constant that depends only on mini=1,...,2s{yi− yi+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.