A degree-at-most- polynomial is a best uniform approximation precisely when its error attains alternating extrema of equal magnitude at at least ordered points.
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The Equioscillation theorem, also known as the Weierstrass Approximation Theorem, is primarily associated with the field of approximation theory, particularly in the context of polynomial approximation of continuous functions. It is most commonly framed in the setting of the uniform approximation of continuous functions on closed intervals.