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The conditions for a best appromation are particularly simple if the function p(''x'') is restricted to polynomials less than a stated degree ''n''.<ref name="powell" /> A theorem by [[Karl Weierstrass]] states that for any function f ∈ [[continuous function|C]][-1,1] and any ''ε'' > 0 then there exists a polynomial p such that<ref name="phillips" />
::<math>\max_{-1 \leq x \leq 1}|f(x)-p(x)| < \epsilon
Polynomial expansions such as the [[Taylor series]] expansion are often convenient for theoretical work but less useful for practical applications. For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of terms in an effort to reduce computational expense of repeated evaluation.
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