On a limit value of a remainder of the Lebesgue constant corresponding to the Lagrange trigonometrical polynomial

The behavior of Lebesgue constant of a trigonometrical Lagrange polynomial interpolating the periodic function in an odd number of clusters is studied. The limit value of the remainder in the known asymptotic formula for this constant is found. A special representation of a remainder allowed us to establish its strict decreasing. On this basis, for a Lebesgue constant, a non-improvable uniform bilateral logarithmic function estimate is received. The extremum problems related to the best approximation of a constant of Lebesgue are solved: quite particular elements of the best approximation and the value of the best approximation are specified.