Coconvex Approximation of Periodic Functions

Let C~ be the space of continuous 2 π-periodic functions f, endowed with the uniform norm ‖ f‖ : = max _x∈R| f(x) | , and denote by ω_k(f, t) , the k-th modulus of smoothness of f. Denote by C~ ^r, the subspace of r times continuously differentiable functions f∈ C~ , and let T_n, be the set of trigonometric polynomials T_n of degree ≤ n (that is, of order ≤ 2 n+ 1). Given a set Ys:={yi}i=12s, of 2s points, s≥ 1 , such that - π≤ y₁< y₂< ⋯ < y_2s< π, and a function f∈ C~ ^r, r≥ 3 , that changes convexity exactly at the points Y_s, namely, the points Y_s are all the inflection points of f. We wish to approximate f by trigonometric polynomials which are coconvex with it, that is, satisfy f′′(x)Tn′′(x)≥0,x∈R.We prove, in particular, that if r≥ 3 , then for every k, s≥ 1 , there exists a sequence {Tn}n=N∞, N= N(r, k, Y_s) , of trigonometric polynomials T_n∈ T_n, coconvex with f, such that ‖f-Tn‖≤c(r,k,s)nrωk(f(r),1/n).It is known that one may not take N independent of Y_s.