Approximation properties of the operators \({\mathcal Y}_{n+2r}(f)\) 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 \(\sum_{k=1}^{2r} a_k P_{n+k}(x)\) 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 \(W_r H_{L_2}^\mu\) 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.
scientific article; zbMATH DE number 1928338