On a convergence of the Fourier-Pade approximation

We consider convergence acceleration of the truncated Fourier series by sequential appli-cation of polynomial and rational corrections. Polynomial corrections are performed alongthe ideas of the Krylov-Lanczos approximation. Rational corrections contain unknown pa-rameters which determination is a crucial problem for realization of the rational approx-imations. We consider approach connected with the Fourier-Pade approximations. Thisrational-trigonometric-polynomial approximation we continue calling the Fourier-Pade ap-proximation. We investigate its convergence for smooth functions in different frameworksand derive the exact constants of asymptotic errors. Detailed analysis and comparisons ofdifferent rational-trigonometric-polynomial approximations are performed and the conver-gence properties of the Fourier-Pade approximation are outlined. In particular, fast conver-gence of the Fourier-Pade approximation is observed in the regions away from the endpoints.