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.
be a continuous function from
Among all the polynomials of degree
, the polynomial
minimizes the uniform norm of the difference
points
) = σ ( − 1
σ
[1][2] 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
being relatively prime polynomials of degree
n − ν
m − μ
, minimizes the uniform norm of the difference
m + n + 2 − min { μ , ν }
points
) = σ ( − 1
σ
[1] Several minimax approximation algorithms are available, the most common being the Remez algorithm.
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