On the approximation of conjugate functions and their derivatives on the segment by partial sums of Fourier - Chebyshev series

In this paper, we study the approximation of conjugate functions with the density $\mathit{f}\in \mathit{H}^{(\alpha)}[-1,1], \alpha\in (0,1],$ on the segment $[-1,1]$ by the conjugate Fourier - Chebyshev series. We establish the order estimations of the approximation depending on the location of a point on the segment. It is noted that approximation at the endpoints of the segment has a higher rate of decrease in comparison with the whole segment. We introduce classes of functions, which, in a certain sense, can be associated with the derivative of a conjugate function on the segment $[-1,1]$, and the approximation of functions from these classes by partial sums of the Fourier - Chebyshev series is studied. An integral representation of the approximation is found. In the case when the density $\mathit{f}\in \mathit{W}^{1}\mathit{H}^{(\alpha)}[-1,1], \alpha\in (0,1],$ the order estimations of the approximation, depending on the location of the point on the segment, are established. The case, when the density $\mathit{f} (t)=|t|^{s}, s>1$, is considered. In this case, an integral representation of the approximation, estimations for pointwise and uniform approximations, as well as an asymptotic estimation for the uniform approximation are obtained. It is noted that the order of the uniform approximations of the function under study by partial sums of the Fourier - Chebyshev series and the corresponding conjugate function by conjugate sums coincide.