On one consequence of the Chebyshev alternance

The classical problem of the best approximation of a continuous function by a polynomial over a Chebyshev system of functions is considered. It is known that the solution of the problem is characterized by alternance. In addition, there is a linear growth function of the deviation of the target function of the coefficients of the polynomial from its minimum value with respect to the deviation of the vector of coefficients from the optimal one. In this article, the formula for the exact coefficient of this linear growth function is obtained by means of convex analysis. In contrast to those obtained earlier, it is expressed in a form constructive for realization through the values of the Chebyshev system functions at the points realizing alternance.