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Equioscillation theorem

From Wikipedia, the free encyclopedia

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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