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Weierstrass Approximation Theorem

If f is a continuous real-valued function on [a,b] and if any epsilon>0 is given, then there exists a polynomial p on [a,b] such that

|f(x)-P(x)|<epsilon

for all x in [a,b]. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.

See also

Müntz's Theorem, Stone-Weierstrass Theorem

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References

Jeffreys, H. and Jeffreys, B. S. "Weierstrass's Theorem on Approximation by Polynomials" and "Extension of Weierstrass's Approximation Theory." §14.08-14.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 446-448, 1988.

Referenced on Wolfram|Alpha

Weierstrass Approximation Theorem

Cite this as:

Weisstein, Eric W. "Weierstrass Approximation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassApproximationTheorem.html

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