Almost coconvex approximation of continuous periodic functions - Dzyubenko, H. A. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
If a 2π -periodic function f continuous on the real axis changes its convexity at 2s, s N, points y_i : π ≤ y_2s < y_2s-1 < . . . < y_1 < π , and, for all other i Z, y_i are periodically defined, then, for every natural n ≥ N_y_i , we determine a trigonometric polynomial P_n of order cn such that P_n has the same convexity as f everywhere except, possibly, small neighborhoods of the points y_i : (y_i _i /n, y_i + π /n), and \| f P_n\| ≤ c(s) ω 4(f, π /n), where N_y_i is a constant depending only on min_i = 1,...,2s{ y_i y_i+1} , c and c(s) are constants depending only on s, ω 4(f, · ) is the fourth modulus of smoothness of the function f, and \| · \| is the max-norm.