The Lagrange trigonometric interpolation polynomial with the minimal norm considered as an operator from $C_{2\pi}$ to $C_{2\pi}$

In this paper we perform a comparative analysis of Lebesgue functions and constants of a family of Lagrange polynomials. We prove that if a polynomial from the family has the minimal norm in the space of square summable functions, then it also has the minimal norm as an operator which maps a space of continuous functions into itself.