Negative results in coconvex approximation of periodic functions

We prove, that for each r∈N, n∈N and s∈N there are a collection {yi}i=12s of points y2s0 is constant, depending only on r. Moreover, we r=0,1,2 any f∈C(r)(R), (−1)i−1f convex [yi,yi−1], 1≤i≤2s, and, sequence {Tn}n=0∞ polynomials Tn, (2), have lim supn→∞nr‖f−Tn‖C(R)ω4(f(r),1∕n)=+∞,where ω4 the fourth modulus continuity.