Abstract
We continue investigations of the modified-trigonometric-rational approximations that arise while accelerating the convergence of the modified Fourier expansions by means of rational corrections. Previously, we investigated the pointwise convergence of the rational approximations away from the endpoints and the $L_2$-convergence on the entire interval. Here, we study the convergence at the endpoints and derive the exact constants for the main terms of asymptotic errors. We show that the Fourier-Pade approximations are much more accurate in all frameworks than the modified expansions for sufficiently smooth functions. Moreover, we consider a simplified version of the rational approximations and explore the optimal values of parameters that lead to better accuracy in the framework of the $L_2$-error. Numerical experiments perform comparisons of the rational approximations with the modified Fourier expansions.