Hermite interpolation on a simplex

In the paper, we solve the problem of polynomial interpolation and approximation functions of several variables on an $n$-dimensional simplex in the uniform norm using polynomials of the third degree. We choose interpolation conditions in terms of derivatives in the directions of the edges of a simplex. In the same terms we obtained estimates of the deviation of derivatives of polynomial from the corresponding derivatives of an interpolated function under the assumption that the interpolated function has continuous directional derivatives up to the fourth order inclusive. We defined a long edge and in these terms we introduce the geometric characteristics of the simplex. It is proved that for dimensions 3 and 4, the interpolation conditions can be chosen so that the estimates the deviations of the derivatives do not depend on the geometry of the simplex, and in the cases of dimensions greater than 4 with the selected interpolation conditions it is impossible.