Theorem

Weierstrass Approximation Theorem

Analysis

Every continuous function on a closed, bounded interval of the real line can be uniformly approximated as closely as desired by a polynomial. Proved by Karl Weierstrass, it underlies much of approximation theory and numerical analysis.

Facts

Statement

Every continuous real-valued function on a closed real interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. 1

Proof Year

1885 1

Classification

Statement Form

Existence Theorem 1

Connections

Sources

1. Weierstrass Approximation Theorem (Wikipedia, Stone-Weierstrass Article)

Wikimedia Foundation

Applications
If f is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that | f (x, y) - p(x, y) | < ε for all x in [a, b] and y in [c, d].
History note on Weierstrass's 1885 proof
The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.

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Weierstrass Approximation Theorem

Has Statement Form

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