Comonotone approximation of periodic functions

Abstract

Let C˜ be the space of continuous 2π-periodic functions f, endowed with the uniform norm ‖f‖≔maxx∈R|f(x)|, and denote by ωm(f,t), the mth modulus of smoothness of f. Denote by C˜r, the subspace of r times continuously differentiable functions f∈C˜, and let Tn, be the set of trigonometric polynomials Tn of degree <n. If f∈C˜r, has 2s, s≥1, extremal points in (−π,π], denote by En(1)(f)≔infTn∈Tn:f′Tn′≥0‖f−Tn‖, the error of its best comonotone approximation. We prove, that if f∈C˜r, then for either m=1, or m=2 and r=2s, or m∈N and r>2s, En(1)(f)≤c(m,r,s)nrωm(f(r),1/n),n≥1,where the constant c(m,r,s) depends only on m, r and s.