simplex

Hermite Interpolation on a Simplex

Authors: Khasyanov R. S.

In the paper, we solve the problem of polynomial interpolation and approximation functions of several variable sonann dimensional simplex in the uniform normus ingpoly nomials of the third degree.Wechoose 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.

Some Properties of 0/1-Simplices

Authors: Nevskii M. V., Ukhalov A. Y.

Let n ∈ N, and let Q n = [0,1] n . For a nondegenerate simplex S ⊂ R n , by σS we mean the homothetic copy of S with center of homothety in the center of gravity of S and ratio of homothety σ. Put ξ(S) = min{σ > 1 : Q n ⊂ σS}, ξ n = min{ξ(S) : S ⊂ Q n }.

On the Geometric Structure of the Continuos Mappings Preserving the Orientation of Simplexes

Authors: Klyachin V. A., Chеbanеnko N. A.

It is easy to show that if a continuous open map preserves the orientation of allsimplexes, the nit is affine. The class of continuous open maps f : D ⊂ R m → R n that preserve the orientation of simplexes from a given subset of a set of simplexes with vertices in the domain D ⊂ R m is considered. In this paper, questions of the geometric structure of linear inverse images of such mappings are studied.

Isoperimetry Coefficient for Simplex in the Problem of Approximation of Derivatives

Authors: Klyachin V. A., Shurkaeva D. V.

We introduce the isoperimetry coefficient σ(G) = |∂G|n/(n−1)/|G| of region G ⊂ Rn. In terms of this the error δΔ(f) estimates for the gradient of the piecewise linear interpolation of functions of class C1(G), C2(G), C1,α(G), 0 < α < 1, are obtained. The problem of obtaining such estimates is nontrivial, especially in the multidimensional case. Here it should be noted that in the two-dimensional case, for functions of class C2(G), the convergence of the derivatives is provided by the classical Delaunay condition.