Home > Analysis > Approximation by polynomials of a function defined on the entire real line

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


Suppose {f: \Bbb{R}\rightarrow \Bbb{R}} is a function which can be uniformly approximated by polynomials on {\Bbb{R}}. Then {f} 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 {f} can be approximated uniformly by a sequence {(P_n)} of polynomials on the entire real line. Then for {n} great enough and {k\geq 1} we have

\displaystyle |P_n(x)-P_{n+k}(x)|\leq |P_n(x)-f(x)|+|P_{n+k}(x)-f(x)|<2\varepsilon

for every {k \geq 1}. But the difference of two polynomials is again a polynomial, and every nonconstant polynomial is unbounded on {\Bbb{R}}. This means that for {n} great enough the polynomials are essentially the same. The only thing that can differ is the free coefficient {c_n}. But since {P_n(0)=c_n} is convergent to {f(0)} it follows that {c_n} is also convergent, and trivially {P_n} uniformly converges to {P_n(x)-P(0)+f(0)}.

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

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

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