Jackson order of approximation by Chebyshev-Fourier series (Q1917167)

\textit{G. Mastroianni} and \textit{J. Szabados} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 33, 375-386 (1994; Zbl 0837.41003)] raised and solved some problems of finding classes of functions where the order of Lagrange interpolation which follows from the global properties of the function can be improved by amending the local properties of the function. The author of the present paper investigates whether the function classes found in the previous problems are as good in the case of approximation by Chebyshev-Fourier series as in the interpolation case. He deals with the continuous case only, but obtains the same conclusion without assuming the continuity of the piecewise second derivative of the function as it was needed in the interpolation case. His first theorem deals with approximation by Chebyshev series of first kind, and the second one with second kind. Using these results he is able to give some estimations in the case of weighted approximation by ultraspherical Fourier series.