Weierstrass Approximation Theorem

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Theorem

Let $f$ be a real function which is continuous on the closed interval $\Bbb I$.

Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy.


Proof

Let $\map f t: \Bbb I = \closedint a b \to \R$ be a continuous function.

Introduce $\map x t$ with a rescaled domain:

$\map f t \mapsto \map x {a + t \paren {b - a} } : \closedint a b \to \closedint 0 1$.

From now on we will work with $x: \closedint 0 1 \to \R$, which is also continuous.

Let $n \in \N$.

For $t \in \closedint 0 1$ consider the Bernstein polynomial:

$\displaystyle \map {B_n x} t = \sum_{k \mathop = 0}^n \map x {\frac k n} \binom n k t^k \paren {1 - t}^{n - k}$

For $t \in \closedint 0 1$, $0 \le k \le n$, let:

$\displaystyle \map {p_{n, k} } t := \binom n k t^k \paren {1 - t}^{n - k}$

By the binomial theorem:

$\displaystyle \sum_{k \mathop = 0}^n \map {p_{n, k} } t = 1$


Lemma 1

$\displaystyle \sum_{k \mathop = 0}^n k \map {p_{n,k} } t = n t$

$\Box$


Lemma 2

$\displaystyle \sum_{k \mathop = 0}^n \paren {k - nt}^2 \map {p_{n,k} } t = n t \paren {1 - t}$

$\Box$


Now we construct the estimates.

\(\displaystyle \sum_{k \mathop : \size {\frac k n \mathop - t} \ge \delta}^n \map {p_{n, k} } t\) \(\le\) \(\displaystyle \sum_{k \mathop : \size {\frac k n \mathop - t} \ge \delta}^n \map {p_{n, k} } t \frac {\paren {k - n t}^2} {\delta^2 n^2}\) $\displaystyle \frac {\paren {k - n t}^2} {\delta^2 n^2} \ge 1$
\(\displaystyle \) \(\le\) \(\displaystyle \frac 1 {n^2 \delta^2} \sum_{k \mathop = 0}^n \paren{k - n t}^2 \map {p_{n, k} } t\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {t \paren {1 - t} } {n \delta^2}\) Lemma 2
\(\displaystyle \) \(\le\) \(\displaystyle \frac 1 {4 n \delta^2}\) $\forall t \in \closedint 0 1: \frac 1 4 \ge t \paren {1 - t}$

Here $\displaystyle \sum_{k \mathop : \size {\frac k n \mathop - t} \ge \delta}^n$ denotes the summation over those values of $k \in \N$, $k \le n$, which satisfy the inequality $\displaystyle \size {\frac k n - t} \ge \delta$.

For some $\delta > 0$ denote:

$\displaystyle \map {\omega_\delta} x := \sup_{\size {t - s} < \delta} \size {\map x s - \map x t}$

Then:

\(\displaystyle \size {\map {B_n x} t - \map x t}\) \(=\) \(\displaystyle \size {\map {B_n x} t - \map x t \sum_{k \mathop = 0}^n \map {p_{n, k} } t}\)
\(\displaystyle \) \(=\) \(\displaystyle \size {\sum_{k \mathop = 0}^n \map x {\frac k n} \map {p_{n, k} } t - \map x t \sum_{k \mathop = 0}^n \map {p_{n, k} } t}\)
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{k \mathop = 0}^n \size {\map x {\frac k n} - \map x t} \map {p_{n, k} } t\) as $\map {p_{n, k} } t \ge 0$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop : \size {\frac k n \mathop - t} < \delta}^n \size {\map x {\frac k n} - \map x t} \map {p_{n, k} } t + \sum_{k \mathop : \size {\frac k n \mathop - t} \ge \delta}^n \size {\map x {\frac k n} - \map x t} \map {p_{n, k} } t\)
\(\displaystyle \) \(\le\) \(\displaystyle \map {\omega_\delta} x \sum_{k \mathop : \size {\frac k n \mathop - t} < \delta}^n \map {p_{n, k} } t + 2 \norm x_\infty \frac 1 {4 n \delta^2}\)
\(\displaystyle \) \(\le\) \(\displaystyle \map {\omega_\delta} x \cdot 1 + \frac {\norm x_\infty} {2 n \delta^2}\)

where $\norm {\,\cdot \,}_\infty$ denotes the supremum norm.


Let $\epsilon > 0$.

By Continuous Function on Closed Interval is Uniformly Continuous, $\map x t$ is uniformly continuous.

We choose $\delta > 0$ such that $\displaystyle \map {\omega_\delta} x < \frac \epsilon 2$.

Choose $\displaystyle n > \frac {\norm x_\infty} {\epsilon \delta^2}$

Then:

$\displaystyle \norm {\map {B_n x} t - \map x t}_\infty < \epsilon$.

$\blacksquare$


Also known as

This result is also seen referred to as Weierstrass's theorem, but as there are a number of results bearing Karl Weierstrass's name, it makes sense to be more specific.


Source of Name

This entry was named for Karl Weierstrass.


Historical Note

The Weierstrass Approximation Theorem has been demonstrated to have far-reaching and important effects in every aspect of the field of analysis.

It has also been given a significant generalisation by Marshall Harvey Stone.


Sources