## Approximation by polynomials of a function defined on the entire real line

Suppose is a function which can be uniformly approximated by polynomials on . Then is also a polynomial.

Note the striking difference between this result and the Weierstrass approximation theorem.

The proof is very simple, and it relies on the basic properties of polynomials. Suppose that can be approximated uniformly by a sequence of polynomials on the entire real line. Then for great enough and we have

for every . But the difference of two polynomials is again a polynomial, and every nonconstant polynomial is unbounded on . This means that for great enough the polynomials are essentially the same. The only thing that can differ is the free coefficient . But since is convergent to it follows that is also convergent, and trivially uniformly converges to .

Since the uniform limit is unique it follows that is a polynomial.

There is a similar result presented here. If a function defined on a compact interval is uniformly approximated by polynomials whose degrees are bounded above, then is also a polynomial.