# 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:

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

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

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

By the binomial theorem:

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

### 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.

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

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

For some $\delta > 0$ denote:

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

Then:

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

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

Let $\epsilon > 0$.

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

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

Then:

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

$\blacksquare$

## 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.