The Weierstrass (Polynomial) Approximation Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The Weierstrass (Polynomial) Approximation Theorem

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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The Weierstrass (Polynomial) Approximation Theorem

Let $f(x)$ be continuous on a real interval $I$. Then for any $\, there exists an $n$th-order polynomial $P_n(f,x)$, where $n$ depends on$\, such that

\

for all $x\.

Thus, any continuous function can be approximated arbitrarily well by means of a polynomial. Furthermore, an infinite-order polynomial can yield an error-free approximation. Of course, to compute the polynomial coefficients using a Taylor series expansion, the function must also be differentiable of all orders throughout $I$.

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