Polynomial Approximations
The Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.
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.
One popular minimax approximation algorithm is the Remez algorithm. Chebyshev polynomials of the first kind closely approximate the minimax polynomial.
Read more about this topic: Minimax Approximation Algorithm