Equioscillation theorem
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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 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.
Algorithms[]
Several minimax approximation algorithms are available, the most common being the Remez algorithm.
References[]
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
Categories:
- Theorems about polynomials
- Numerical analysis
- Theorems in analysis
- Mathematical analysis stubs