Is the polynomial interpolation of infinite points unique?

Question

Is there a rigorous proof of the uniqueness of the polynomial interpolation of an infinite set of points ? Otherwise is there a counterexample ?

Answer 1

In general there is no such polynomial!

Suppose we work over an infinite field $K$. There is no polynomial function $f:K \rightarrow K$ such that $f(0)=1$ and $f(x) = 0$ for any $x \neq 0$. Indeed, a nonzero polynomial function $f:K \rightarrow K$ of degree $d$ has at most $d$ roots in $K$.

Answer 2

As pointed out in the other answer, existence is not guaranteed. But if you do have a polynomial $f(x)$ that interpolates infinitely many given points, that polynomial is unique.

This fact follows from the Fundamental Theorem of Algebra; if $f(x)$ and $g(x)$ are two polynomials interpolating your points, then $f-g$ has infinitely many roots and so must be the zero polynomial.