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We investigate convergence of the rational-trigonometric-polynomial interpolations whichperform convergence acceleration of the classical trigonometric interpolation by sequentialapplication of polynomial and rational corrections. Rational corrections contain unknownparameters which determination outlines the behavior of the interpolations in different frame-works. We consider approach for determination of the unknown parameters by minimizationof the constants of the asymptotic errors. We perform theoretical and numerical analysis ofsuch optimal interpolations.