Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier–Legendre sums of order $n$ with $2r$ terms of the form $\sum _{k=1}^{2r}a_kP_{n+k}(x)$ added; here $P_m(x)$ denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval $[-1,1]$, which, in fact, for $r=1$ allows us to significantly improve the approximation properties of partial Fourier–Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions $W_rH_{L_2}^\mu $ and $A_q(B)$. With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.