About the only solution in the problem of the best plural reflection's approximation by algebraic polynomial

This paper is devoted to the proof of the theorem including necessary and sufficient conditions in the problem of the best plural reflection's approximation by algebraic polynomial. In the proof is used several author's were published results and two auxiliary lemmas. The proof is based on the minim ax's problems theory, the approximation's theory by algebraic polynomials of the P.L. Chebyshev and the plural's analysis.