Equioscillation theorem

In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev.[1]

Statement

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Let   be a continuous function from   to  . Among all the polynomials of degree  , the polynomial   minimizes the uniform norm of the difference   if and only if there are   points   such that   where   is either -1 or +1.[1][2]

Variants

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The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree   and denominator has degree  , the rational function  , with   and   being relatively prime polynomials of degree   and  , minimizes the uniform norm of the difference   if and only if there are   points   such that   where   is either -1 or +1.[1]

Algorithms

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Several minimax approximation algorithms are available, the most common being the Remez algorithm.

References

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  1. ^ a b c Golomb, Michael (1962). Lectures on Theory of Approximation.
  2. ^ "Notes on how to prove Chebyshev's equioscillation theorem" (PDF). Archived from the original (PDF) on 2 July 2011. Retrieved 2022-04-22.
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