Cardinal-polynomial interpolation

Let $n\in \mathbb{N}$ with $n\ge 0$, let $x_0<\cdots <x_n$ be distinct real nodes, and let $y_0,\dots ,y_n\in \mathbb{R}$. Define the node polynomial and barycentric weights by ${\omega }_n(X)={\prod }_{k=0}^n(X-x_k),w_j={({\prod }_{\begin{array}{c}0\le k\le n\\ k\mathrm{\ne }j\end{array}}(x_j-x_k))}^{-1}(0\le j\le n)\mathrm{.}$ An empty product is $1$, so $w_0=1$ when $n=0$.

In these formulas, $X$ is a real argument, and for real $a$ and nonzero real $b$, $a\mathrm{/}b$ denotes $a{b}^{-1}$ in the field $\mathbb{R}$ (Field).

For each $j$, the cardinal function is ${\mathrm{\ell }}_j(X)={\prod }_{\begin{array}{c}0\le k\le n\\ k\mathrm{\ne }j\end{array}}\frac{X-x_k}{x_j-x_k}\mathrm{.}$ The interpolation function for the data $y=(y_0,\dots ,y_n)$ is $I_{n}y(X)={\sum }_{j=0}^n{y}_j{\mathrm{\ell }}_j(X)\mathrm{.}$ For a real function $f$ defined at the nodes, write $I_{n}f:=I_n(f(x_0),\dots ,f(x_n))$.

For products indexed by $k\mathrm{\ne }j$, use the length-$n$ list ${\widehat{x}}_{j,k}$ ($0\le k<n$), where ${\widehat{x}}_{j,k}=x_k$ for $k<j$ and ${\widehat{x}}_{j,k}=x_{k+1}$ for $k\ge j$; the product is in this order and is empty when $n=0$. Strict ordering gives $x_j-{\widehat{x}}_{j,k}\mathrm{\ne }0$ for every factor, so A finite product of nonzero reals is nonzero and the field inverse law make each $w_j$ well defined.