On a rational linear approximation of Fourier series for smooth functions (Q2492777)

The topic of the paper is to improve the convergence of the Fourier series, a branch of numerical analysis. It is stated that the Fourier series of functions of \(p\)-derivatives \((p\geq 0)\) with discontinuities have slow convergence on the approximation interval because of the Gibbs phenomena. Referring to many works on this subject, the authors would propose a class of Fourier-Padé approximation to ``eliminate'' the Gibbs phenomena [see \textit{T. A. Driscoll} and \textit{B. Fornberg}, Numer. Algorithms 26, No. 1, 77--92 (2001; Zbl 0973.65133)]. A theorem developing convergence degree is proved. Numerical examples are given. There is a lot of bibliography.