Geodesic
On two-dimensional polynomial interpolation
Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 211-223 Cet article a éte moissonné depuis la source Math-Net.Ru

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A tuple $\mathfrak{N}=\{n_1,\,n_k;\,n\}$ of positive integers with $\sum_{\nu=1}^k n_\nu(n_\nu+1)=(n+1)(n+2)$ is said to be regular if there exists a set $U=\{u_1,\,\dots,\,u_k\}\subset\mathbb{R}^2$ such that the Hermite interpolation problem $(\mathfrak{N},\,U)$ is regular, i.e., for arbitrary numbers $\lambda_{(i,j),\nu}$, $i+j, $\nu=1,\dots,k$, there exists a unique polynomial $P(x,\,y)\in\pi_n(\mathbb{R}^2)$ such that $$ {\partial^{i+j}\over\partial x^i\partial y^j}P(x,y)\big|_{u_\nu}=\lambda_{(i,j),\nu},\qquad i+j<n_\nu,\quad \nu=1,\dots,k. $$ In this paper an algorithm is obtained that completely describes the regular and singular tuples $\mathfrak{N}$ under the condition that $n_{10}=1$. In the case when only the derivatives of order $n_\nu$ are interpolated, necessary and sufficient conditions are obtained for an arbitrary tuple $\mathfrak{N}$ to be regular. 
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A. A. Akopian; O. V. Gevorgyan; A. A. Sahakian. On two-dimensional polynomial interpolation. Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 211-223. http://geodesic.mathdoc.fr/item/SM_1993_76_1_a11/

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