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Zbl 0951.65008
Gasca, Mariano; Sauer, Thomas
On bivariate Hermite interpolation with minimal degree polynomials.
(English)
[J] SIAM J. Numer. Anal. 37, No.3, 772-798 (2000). ISSN 0036-1429; ISSN 1095-7170/e

Let $r= ax+ by+ c$ be a polynomial for which $a^2+ b^2= 1$, $a> 0$, or $a= 0$ and $b> 0$. Both the polynomial $r$ and the straight line $r= 0$ are denoted by the same symbol. Let $\Gamma= \{r_0,\dots, r_n\}$, $\Gamma'= \{r_0',\dots, r_m'\}$ be two systems of straight lines in $\bbfR^2$, such that each pair $(r_i, r_j')\in \Gamma\times \Gamma'$ intersects at exactly one point $u_{ij}\in \bbfR^2$. Let $I= \{(i,j)/0\le i\le n, 0\le j\le m(i)\}$, where $m(i)\ge m(i+ 1)$, and the points $u_{ij}$, $(i,j)\in I$, are the intersections of only one pair $(r_i, r_j')$. The Newton basis $B_S$, $S= (\Gamma\times \Gamma', I)$, is defined as $B_S= \{\phi_{ij}/(i,j)\in I\}$, where $\phi_{ij}= \prod^{i- 1}_{h= 0} r_h \prod^{j- 1}_{k= 0} r_k'$. The interpolation data are defined by $$L_{ij}f= {\partial^{s_i+ t_j} f\over\partial \rho^{\prime s_i}_j\partial \rho^{\prime t_j}_i} (u_{ij}),\tag 1$$ where $s_i$ (resp. $t_j$) is the number of lines $r_h(r_k')$ with $h< i$ $(k< j)$, which coincide with $r_i(r_j')$. Using the above notation, the paper states the interpolation problem as follows: for a given set $\{z_{ij}\mid (i,j)\in I\}\subset \bbfR$, find a polynomial $p\in V_S= [B_S]$, such that $L_{ij}p= z_{ij}$, for every $(i,j)\in I$. The existence of the unique solution $p$ of the form $p= \sum_{(i,j)\in I} a_{ij}\phi_{ij}$, for this Hermite interpolation problem is proved.\par Some other questions are investigated on this problem. It is proved that the interpolation operator (1) is degree-reducing, and the interpolation space is a minimal degree space for the problem. By means of divided differences the solution of the problem is constructed recursively, and a remainder formula for this problem is obtained using a finite difference technique.
[Jesus Illán González (Vigo)]
MSC 2000:
*65D05 Interpolation (numerical methods)
65D10 Smoothing
41A05 Interpolation
41A10 Approximation by polynomials
41A63 Multidimensional approximation problems

Keywords: bivariate Hermite interpolation; minimal degree polynomials; polynomial interpolation; reversible systems; error representation; degree reduction; divided difference method

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