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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.

Statement

Let ${\displaystyle f}$ be a continuous function from ${\displaystyle }$ to ${\displaystyle \mathbb {R} }$. Among all the polynomials of degree ${\displaystyle \leq n}$, the polynomial ${\displaystyle g}$ minimizes the uniform norm of the difference ${\displaystyle \|f-g\|_{\infty }}$ if and only if there are ${\displaystyle n+2}$ points ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }}$ where ${\displaystyle \sigma }$ is either -1 or +1.

Variants

The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree ${\displaystyle \leq n}$ and denominator has degree ${\displaystyle \leq m}$, the rational function ${\displaystyle g=p/q}$, with ${\displaystyle p}$ and ${\displaystyle q}$ being relatively prime polynomials of degree ${\displaystyle n-\nu }$ and ${\displaystyle m-\mu }$, minimizes the uniform norm of the difference ${\displaystyle \|f-g\|_{\infty }}$ if and only if there are ${\displaystyle m+n+2-\min\{\mu ,\nu \}}$ points ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }}$ where ${\displaystyle \sigma }$ is either -1 or +1.

Algorithms

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

References

1. ^ 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.

External links

• Notes on how to prove Chebyshev’s equioscillation theorem at the Wayback Machine (archived July 2, 2011)
• The Chebyshev Equioscillation Theorem by Robert Mayans
• The de la Vallée-Poussin alternation theorem at the Encyclopedia of Mathematics
• Approximation theory by Remco Bloemen

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