Suppose I have a set of $n$ points on the plane at $(x_i, y_i)$, and I want to construct the polynomial interpolating them. This can be done with Lagrange interpolation in the straightforward manner.
Now suppose I add two points at $(a,0)$ and $(b,0)$, where $a < \min(x_i)$ and $b > \max(x_i)$. In other words, I add two zeros, one on either side of the set of points. We can again perform Lagrange interpolation on this augmented set of points to obtain a polynomial that approaches 0 as you approach $a$ and $b$.
This defines a family of polynomials for each choice of $a$ and $b$. I suspect this ought to converge pointwise to some well-defined function (though I haven't proven this).
Does this converge pointwise to something, and if so, what?
In effect we end up with Lagrange interpolation, but with extra zeros "at infinity." Does this lead to a well-defined interpolation method?
If we write the Lagrange polynomials for the set of points including $a$ and $b$ as $L^{a,b}_j(x)$ and $L_j(x)$ for those without, then they are given by $$ L^{a,b}_j(x) = \frac{(x-a)(x-b)}{(x_j-a)(x_j-b)}\prod_{i \neq j} \frac{x-x_i}{x_j-x_i} = \frac{(x-a)(x-b)}{(x_j-a)(x_j-b)} L_j(x). \tag{*} $$ Now, since the function values at $a$ and $b$ are zero, the interpolating polynomial is $$ \sum_j y_j L_j^{a,b}(x), $$ not including $L_a$ or $L_b$. Therefore, every term is of the form in (*), and simple results about rational functions tell us that for fixed $x$, $$\lim_{\substack{a \to -\infty \\ b \to \infty}} \frac{(x-a)(x-b)}{(x_j-a)(x_j-b)} = 1, $$ so $L^{a,b}_j(x) \to L_j(x)$ pointwise.