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Statement

Let f {\displaystyle f} be a continuous function from [ a , b ] {\displaystyle [a,b]} to R {\displaystyle \mathbb {R} } . Among all the polynomials of degree ≤ n {\displaystyle \leq n} , the polynomial g {\displaystyle g} minimizes the uniform norm of the difference ‖ f − g ‖ ∞ {\displaystyle \|f-g\|_{\infty }} if and only if there are n + 2 {\displaystyle n+2} points a ≤ x 0 < x 1 < ⋯ < x n + 1 ≤ b {\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b} such that f ( x i ) − g ( x i ) = σ ( − 1 ) i ‖ f − g ‖ ∞ {\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }} where σ {\displaystyle \sigma } is either -1 or +1.²³

Variants

The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree ≤ n {\displaystyle \leq n} and denominator has degree ≤ m {\displaystyle \leq m} , the rational function g = p / q {\displaystyle g=p/q} , with p {\displaystyle p} and q {\displaystyle q} being relatively prime polynomials of degree n − ν {\displaystyle n-\nu } and m − μ {\displaystyle m-\mu } , minimizes the uniform norm of the difference ‖ f − g ‖ ∞ {\displaystyle \|f-g\|_{\infty }} if and only if there are m + n + 2 − min { μ , ν } {\displaystyle m+n+2-\min\{\mu ,\nu \}} points a ≤ x 0 < x 1 < ⋯ < x m + n + 1 − min { μ , ν } ≤ b {\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{m+n+1-\min\{\mu ,\nu \}}\leq b} such that f ( x i ) − g ( x i ) = σ ( − 1 ) i ‖ f − g ‖ ∞ {\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }} where σ {\displaystyle \sigma } is either -1 or +1.⁴

Algorithms

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

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

References

1. Golomb, Michael (1962). Lectures on Theory of Approximation. https://books.google.com/books?id=DbnPAAAAMAAJ ↩

2. Golomb, Michael (1962). Lectures on Theory of Approximation. https://books.google.com/books?id=DbnPAAAAMAAJ ↩

3. "Notes on how to prove Chebyshev's equioscillation theorem" (PDF). Archived from the original (PDF) on 2 July 2011. Retrieved 2022-04-22. https://web.archive.org/web/20110702221651/http://www.math.uiowa.edu/~jeichhol/qual%20prep/Notes/cheb-equiosc-thm_2007.pdf ↩

4. Golomb, Michael (1962). Lectures on Theory of Approximation. https://books.google.com/books?id=DbnPAAAAMAAJ ↩