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]
Let
f
[a,b]
R
\len
g
\|f-g\|infty
n+2
a\lex0<x1< … <xn+1\leb
f(xi)-g(xi)=\sigma(-1)i\|f-g\|infty
\sigma
The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree
\len
\lem
g=p/q
p
q
n-\nu
m-\mu
\|f-g\|infty
m+n+2-min\{\mu,\nu\}
a\lex0<x1< … <xn+1\leb
f(xi)-g(xi)=\sigma(-1)i\|f-g\|infty
\sigma
Several minimax approximation algorithms are available, the most common being the Remez algorithm.