Almost Coconvex Approximation of Continuous Periodic Functions

If a 2.-periodic function f continuous on the real axis changes its convexity at 2s, s 2 N, inflection points yi : -.. y2s < y2s- 1 <... < y1 <. and, for all other i 2 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 -./n, yi +./n) and, moreover, parallel to f - P-n parallel to <= c(s) !4(f,./n), where Nyi is a constant that depends only on mini= 1,...,2s{yi - yi+1}, c and c(s) are constants that depend only on s, !4(f, center dot) is the fourth modulus of smoothness of the function f, and k center dot k is the max-norm.