On the constants in the inverse theorems for the first derivative

The known proofs of the inverse theorems of the theory of approximation by trigonometric polynomials and by functions of exponential type are based on the idea of S. N. Bernstein to expand a function in a series containing its functions of best approximation. In this paper, a new method to establish the inverse theorems is introduced. We establish simple identities that immediately imply the inverse theorems mentioned and, moreover, with better constants. This method can be applied to derivatives of arbitrary order (not necessarily an integer one) and (with certain modifications) to estimates of some other functionals in terms of best approximations. In this paper, the case of the first derivative of a function itself and of its trigonometrically conjugate is considered.