On the existence of trigonometric Padé approximations

In this work, based on the well-known results on classical Padé approximants of power series, the conditions are found under which trigonometric Padé – Jacobi approximants exist for a given Fourier series. This made it possible to describe the class of Fourier series in Chebyshev polynomials of the first and second kind, for which there are nonlinear Padé – Chebyshev approximants. In particular, another proof of the well-known theorem of S.P. Suetin is given.