On the constants in the inverse theorems for the norms of derivatives

We propose a new method for proving the classical inverse theorems of the theory of approximation by trigonometric polynomials and entire functions of exponential type. The method is based on the construction of identities expressing the derivatives of a function itself and of its trigonometrical conjugate in terms of convolution operators. As a consequence, we reduce the constants in the estimates of the norms of the derivatives in terms of best approximations.