Existence and uniqueness of consistent Hermite – Fourier approximations

The trigonometric analogues of algebraic Hermite – Padé approximations were defined, these are Hermite – Fourier approximations. In particular, the theorem of existence of Hermite – Fourier approximations was proved, the sufficient condition of their uniqueness was obtained, and the criterion of the existence and uniqueness of Hermite – Fourier polynomials, which are the numerator and denominator of Hermite – Fourier approximations associated with an arbitrary set of trigonometric series k. When the conditions of the criterion were met, the explicit type of the specified polynomials was established.