ON MULTIVARIATE SEGMENTAL INTERPOLATION PROBLEM

Abstract

In this paper the following problem is introduced, which we call segmental interpolation problem, or briefly segmental problem: Suppose $\chi = \{ \textrm{x}^{(\nu)} : \nu\in I \}$ is a finite or infinite set of knots in $\mathbb{R}^d$. Suppose also that $\mathcal{S}_I = \{ [\alpha_{\nu},\beta_{\nu}] : \nu\in I \}$ is a respective set of any segments. The segmental problem $\{ \chi, S \}$ is to find a polynomial $p$ in $d$ variables and of total degree less than or equal to $n$, satisfying the conditions \[\alpha_{\nu} \le p(\textrm{x}^{(\nu)})\le\beta_{\nu},\quad \forall \nu\in I.\] We bring a necessary and sufficient condition for the solvability of the segmental problem. In case when the problem is solvable and the set of knots $\chi_I$ is finite, we bring a method to find a solution of the segmental problem.