Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients

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.
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