Equioscillation

Equioscillation is a property of a function in which its values oscillate between two extreme bounds with equal magnitude, often in the context of approximation errors. In this setting, the error between a target function and an approximant reaches the same maximum absolute value, E, at a sequence of points within the domain, with the signs of the error alternating.

In approximation theory, equioscillation is central to the minimax or best uniform approximation. The alternation theorem states that, for a continuous function on an interval, there exists a polynomial of degree at most n such that the maximum deviation from the function is minimized, and the resulting error function attains its maximum absolute value at least n+2 times, with alternating signs. If another approximant achieves the same maximum error, its set of alternation points must interleave with that of the first.

A classic illustration is the Chebyshev polynomials T_n on [-1,1], which oscillate between -1 and 1 at n+1 points. This extremal behavior helps drive minimax approximations and explains why Chebyshev polynomials frequently arise in numerical methods for polynomial approximation.

Equioscillation also generalizes beyond polynomials to rational, spline, and other function classes, and it underpins many algorithms in numerical analysis, filter design, and data fitting that seek uniform accuracy rather than pointwise error minimization.