Copositive approximation of periodic functions

Abstract

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and

$$ \left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right), $$

where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₃(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.