Geodesic
Trigonometric Pad\'e approximants for functions with regularly
Sbornik. Mathematics, Tome 200 (2009) no. 7, pp. 1051-1074

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Sufficient conditions describing the regular decrease of the coefficients of a Fourier series $f(x)=a_0/2+\sum a_n\cos{kx}$ are found which ensure that the trigonometric Padé approximants $\pi^t_{n,m}(x;f)$ converge to the function $f$ in the uniform norm at a rate which coincides asymptotically with the highest possible one. The results obtained are applied to problems dealing with finding sharp constants for rational approximations. Bibliography: 31 titles.
Keywords: Fourier series, trigonometric Padé approximants, Padé-Chebyshev approximants, best rational approximations. 
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     author = {Yu. A. Labych and A. P. Starovoitov},
     title = {Trigonometric {Pad\'e} approximants for functions with regularly},
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Yu. A. Labych; A. P. Starovoitov. Trigonometric Pad\'e approximants for functions with regularly. Sbornik. Mathematics, Tome 200 (2009) no. 7, pp. 1051-1074. http://geodesic.mathdoc.fr/item/SM_2009_200_7_a2/