Weierstrass Approximation Theorem
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
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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
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$)
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Weierstrass approximation theorem
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Weierstrass's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Weierstrass's theorem
