interpolation points $x^1,\cdots x^N$.

Note that at some stage of the construction we may have a multiindex $\gamma$ such that $\gamma-\alpha=1_{i_0}$ for some $\alpha$ and some index $i_0$. Then the terms in the $\alpha$th derivative of the eikonal equation evaluated at $x^k$ that do not annihilate a term of form $c_{\gamma}^k\Pi_{l=0}^{k-1}(x-x^l)^{\gamma+{\bf 1}}(x-x^k)^{\gamma}$ are quite easily computed. For this reason the constants of the form $c_{\gamma}^k$ are quite easily computed. You can see very easily this by writing the $\alpha$th derivative of the eikonal equation invoking symmetry $a_{ij}=a_{ji}$. We have $$\label{alpha}
\begin{array}{ll}
\partial^{\alpha}d^2(x,y)=\partial^{\alpha}\left(\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial d^2}{\partial x_i}\frac{\partial d^2}{\partial x_j} \right)\\
\\
=\frac{1}{2}\sum_{ij}a_{ij}(x)\left( \partial^{\alpha}\frac{\partial d^2}{\partial x_i}\right) \frac{\partial d^2}{\partial x_j} +\frac{1}{4}\sum_{ij}\left(\frac{\partial^{\alpha}}{\partial x^{\alpha}} a_{ij}(x)\right) \frac{\partial d^2}{\partial x_i} \frac{\partial d^2}{\partial x_j}\\
\\
 +\frac{1}{4}\sum_{ij}\sum_{\beta <\alpha}\sum_{\gamma \leq\beta}\binom{\alpha}{\beta}\binom{\beta}{\gamma}
\left( \partial^{\beta}a_{ij}(x)\right) 
 \left( \partial^{\alpha -\beta-\gamma}\frac{\partial d^2}{\partial x_i}\right)  \partial^{\gamma} \frac{\partial d^2}{\partial x_j} 
\end{array}$$ If the indicated approximation is plugged into and evaluated at $x^k$ only the terms $\frac{1}{2}\sum_{j}a_{i_0j}(x)\left( \partial^{\alpha}\frac{\partial d^2}{\partial x_{i_0}}\right) \frac{\partial d^2}{\partial x_j}$ (evaluated for approximations $d^2_{\gamma k}$ at interpolation point $x^k$) do not annihilate terms of form $c_{\gamma}^k\Pi_{l=0}^{k-1}(x-x^l)^{\gamma+{\bf 1}}(x-x^k)^{\gamma}$.

Error estimates for the regular polynomial interpolation algorithm
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We first consider error estimates for $L^p$-approximations, and then extend our estimates to $H^{s,p}$-approximations. In the whole Section we consider a bounded domain 