Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials

Abstract: For trigonometric series and series of Chebyshev polynomials, we defined trigonometric Hermite-Pad\'e and Hermite-Jacobi approximations, linear and nonlinear Hermite-Chebyshev approximations. We established criterion of the existence and uniqueness of trigonometric Hermite-Pad\'e polynomials, associated with an arbitrary set of $k$ trigonometric series, and we found explicit form of these polynomials. Similar results were obtained for linear Hermite-Chebyshev approximations. We made examples of systems of functions for which trigonometrical Hermite-Jacobi approximations are existed but aren't the same as trigonometric Hermite-Pad\'e approximations. Similar examples were made for linear and nonlinear Hermite-Chebyshev approximations.