The correctness problem for best approximations by trigonometric polynomials in the class W o r H[?

Abstract

Suppose thatk, rez+, W o r H[ω]C= {f∶f is a 2π-periodic function,fe Cr [−π, π], ω (f(r), δ) ⩽ω (δ)}, Tk is the space of trigonometric polynomials of order k, pk(f)eTk is the polynomial of best uniform approximation to f, and Ek(f) is the error of the best approximation. It is shown that for an arbitrary e > 0 we have , where for 0 0.R (e) is the root of the equation $$R = (\varepsilon ^l R)^{r^l (2k)} \omega ((\varepsilon \cdot R)^{1(2k)} )$$ , and for k = 0 or e > ω(1) we have R(e)=e.