Extremal Trigonometrical and Power Polynomials of Several Variables 0

We consider the set σP of the power non-negative polynomials of several variables.By QP we denote the class of the polynomials from σ1 which can be represented as a sum of squares.It is shown in the classic work by D.Hilbert[3] that QP does not coincide with σP .Step by step a number of polynomials belonging to σP but not belonging to QP was constructed(see[4]-[6]).It is interesting to note that many of these polynomials turn to be extremal in the class σP [2]. In our paper we have made an attempt to work out a general approach to the investigation of the extremal elements of the convex sets QP and σP .It seems to us that we have achieved a considerable progress in the case of QP .In the case of σP we have made only the first steps.We also consider the class σR of the non-negative rational functions. The article is based on the following methods: 1.We investigate non-negative trigonometrical polynomials and then with the help of the Calderon transformation we proceed to the power polynomials. 2.The way of constructing support hyperplanes to the convex sets QP and σP is given in the paper. Now we start with a more detailed description of the results of this article. Let us denote by σ(N1, N2, N3) the set of the trigonometrical polynomials