Comonotone approximation by hybrid polynomials

Abstract

Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R⁡|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree <n. We call Qn(x):=Tn(x)+ax, a∈R, a hybrid polynomial.If f is monotone, then g was called, by Salem and Zygmund, of monotone type.If f has an even number of extremal points in (−π,π], then we estimateinf⁡{‖f−Qn‖:Qns.t.f′(x)Qn′(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials.We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.