The Weierstrass (Polynomial) Approximation Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Weierstrass (Polynomial) Approximation Theorem

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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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