 Lagrange cardinal polynomial

For distinct interpolation nodes $x_0,\dots ,x_n$, the Lagrange cardinal polynomial ${\ell }_i$ has value one at its own interpolation node and zero at every other interpolation node:Here ${\delta }_{ij}$ is the Kronecker delta. These Lagrange cardinal polynomials form a basis of the vector space of polynomials of polynomial degree at most $n$: every such polynomial has the unique expansion $p(x)={\sum }_{i}p(x_i){\ell }_i(x)$. The nodal evaluation linear functionals $E_j(p)=p(x_j)$ form the dual basis, since $E_j({\ell }_i)={\delta }_{ij}$.

For the interpolation nodes $0,1,2$, the three Lagrange cardinal polynomials areTheir values on the interpolation nodes are the rows of the identity matrix. In the figure, each colored curve passes through its own unit-height marker and through zero at the other two interpolation nodes. Multiplying the curves by the data $1,2,5$ and adding gives the Lagrange polynomial $x^2+1$.

Since the constant polynomial one interpolates data values all equal to one, uniqueness gives ${\sum }_i{\ell }_i(x)=1$. Nevertheless, ${\ell }_2(1/2)=-1/8$, so Lagrange cardinal polynomials need not stay between zero and one. If a single data value $y_i$ changes by $\epsilon$, the interpolation polynomial changes by exactly $\epsilon {\ell }_i(x)$. Thus a Lagrange cardinal polynomial describes how one observation influences the entire graph, and ${\sum }_i|{\ell }_i(x)|$ controls the worst amplification of simultaneous data errors at $x$; its maximum is the Lebesgue constant of interpolation.