Interpolant ($g(x_i)=y_i$)

An interpolant is a function satisfying the prescribed sample conditions $g(x_i)=y_i$. It agrees with the data exactly, but need not agree with an unknown underlying function between samples. The polynomial interpolation example shows that a family or a polynomial degree restriction can supply uniqueness; the McShane extension theorem shows a useful nonunique alternative.

A rational interpolant is a rational function agreeing with given data at every interpolation node. If the representation is required to have no zero denominator at those nodes, this condition is part of the problem, in addition to the degree restrictions. Even when removable common factors are cancelled before evaluation, the resulting rational function must still take the prescribed values. For example, at the data $(0,0),(1,1)$ there is no interpolant $a/(x-b)$: the first value forces $a=0$ when its denominator is nonzero, and then the second value cannot be one. The cleared equations nevertheless have the unique coefficient solution $a=0,b=1$.