WOLFRAM NOTEBOOK

Weierstrass Approximation Theorem

The Weierstrass approximation theorem states that polynomials are dense in the set of continuous functions. More explicitly, given a positive number
ϵ
and a continuous real-valued function
f(x)
defined on
[0,1]
, there is a polynomial
p(x)
such that
|f(x)-p(x)
|
∞
<ϵ
. Here
|·
|
∞
is the infinity (or supremum) norm, which in this case (because the closed unit interval is compact) can be taken to be the maximum. The proof of the theorem is based on Bernstein polynomials constructed from
f(x)
,
B
n
(f)(x)
.
This Demonstration provides four different functions to which the theorem is applied. Choose between the four functions and observe how the Bernstein polynomial approximates the function chosen according to the degree stated for
B
n
(f)(x)
. Then click the "difference" button and observe the graph of the absolute value of the difference between the function chosen and the corresponding Bernstein polynomial.

External Links

Permanent Citation

Fabián Alejandro Romero, Michael Ford
​
​"Weierstrass Approximation Theorem"​
​http://demonstrations.wolfram.com/WeierstrassApproximationTheorem/​
​Wolfram Demonstrations Project​
​Published: January 12, 2015
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