On a convergence of the Fourier-Pade approximation
We consider convergence acceleration of the truncated Fourier series by sequential application of polynomial and rational corrections. Polynomial corrections are performed along the ideas of the Krylov-Lanczos approximation. Rational corrections contain unknown parameters which determination is a crucial problem for realization of the rational approximations. We consider approach connected with the Fourier-Pade approximations. This rational-trigonometric-polynomial approximation we continue calling the Fourier-Pade approximation. We investigate its convergence for smooth functions in different frameworks and derive the exact constants of asymptotic errors. Detailed analysis and comparisons of different rational-trigonometric-polynomial approximations are performed and the convergence properties of the Fourier-Pade approximation are outlined. In particular, fast convergence of the Fourier-Pade approximation is observed in the regions away from the endpoints.