Equioscillation theorem

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.

Let ${\displaystyle f}$ be a continuous function from ${\displaystyle [a,b]}$ to ${\displaystyle \mathbf {R} }$. 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 ${\displaystyle \sigma =\pm 1}$.

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