A complete description of the Lebesgue functions for classical Lagrange interpolation polynomials

No explicit forms of Lebesgue functions were described in the mathematical literature by now. This issue is related to the problem of reducing the sum of a finite number of modules of fundamental polynomials or the corresponding Dirichlet kernels. That is why the complete study of graphs of functions for well-known forms remained a complex topical problem in the theory of approximation of functions. All these problems are completely solved in this paper both for an odd number of interpolation nodes and for an even one. To this end we apply elements of the differential calculus to various explicit forms of Lebesgue functions; the mentioned forms were obtained for the first time.