On family of interpolated polynomials of some variables

The one-parametrical family of quadratic interpolated polynomials of several variables is investigated. In a role of parameter the point of $n$-dimensional space acts. Questions of existence and uniqueness interpolated polynomials are investigated. For polynomials the obvious representation (in barycentric system of coordinates) is proved. It is shown that only for the unique parameter continuous docking of interpolated polynomials constructed on elements of a triangulation of a special type takes place. For interpolated polynomial appropriating the given parameter the obvious representation in the Cartesian system of coordinates is proved. Application of interpolation with the given parameter makes possible quadratic spline-approximation of functions of many variables (at the same time with approximation of a field of a gradient of this function).