On the existence of trigonometric Hermite – Jacobi approximations and non-linear Hermite – Chebyshev approximations

In this paper, analogues of algebraic Hermite – Padé approximations are defined, being trigonometric Hermite – Padé approximations and Hermite – Jacobi approximations. Examples of functions are represented for which trigonometric Hermite – Jacobi approximations exist but are not the same as trigonometric Hermite – Padé approximations. Similar examples are made for linear and non-linear Hermite – Chebyshev approximations, which are multiple analogues of linear and non-linear Padé – Chebyshev approximations. Each type of examples follows from the well-known representations for the numerator and denominator of fractions, introduced by C. Hermite when proving the transcendence of number $e$.