On constants in abstract inverse theorems of approximation theory

In classical inverse theorems of the constructive theory of functions, structure characteristics of a function are described in terms of the rate of its approximation. Mostly, proofs of the invers theorem are based on Bernstein's idea of expanding a function in a series involving the polynomials of best approximation for this finction. In the present paper, Bernstein's approach is modified by replasing sums with integrals. It turns out that the inequalities are then deduced form identities of Frullani integrals type. The arguments are of fairly general nature, which makes it possible to obtain analogs of inverse theorems for functionals on abstract Banach (and even quasi-normed) spaces. Abstract results are used to deduce inverse theorems in specific function spaces, including weighted spaces, with specific constants.