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.
On , the chord of is . Its nonnegative gap has maximum at , as differentiation shows. Shifting the chord upward by gives error at . The Chebyshev alternation theorem therefore proves that is the unique best linear uniform approximation, with error . Endpoint differentiability is unnecessary.
Let a polynomial of degree at most have errors at ordered distinct points. Then bounds below the error of the best uniform approximation. Otherwise an improved polynomial with error strictly below would make alternate strictly in sign at all those points, giving at least distinct roots of a polynomial by the intermediate value theorem. Its degree is at most , a contradiction. Moreover, if is not optimal, . In the equality case, an optimal gives weak alternating signs of ; its vanishing order- divided difference has same-sign summands, forcing every nodal value to vanish and hence .
For real with positive st derivative, equality of consecutive best uniform approximation errors would make a best degree-at-most- polynomial also best at degree . The Chebyshev alternation theorem then supplies alternating extrema of its nonzero error, hence distinct zeros. Repeated Rolle's theorem forces a zero of the st derivative of that error, contradicting the assumed sign. The same conclusion holds for a strictly negative derivative by replacing with .
For real continuous -periodic , a degree-at-most- real trigonometric polynomial is its unique best uniform approximation exactly when has alternating extrema of magnitude at distinct ordered points in one period. The closing interval around the circle also counts. A strictly improving approximant would force its difference from to have at least zeros, whereas a nonzero degree- trigonometric polynomial has at most .
For an odd integer and positive summable coefficients , the Weierstrass M-test makes continuous. Its partial sum over is the unique best uniform approximation by degree-at-most- trigonometric polynomials, for . If is the first omitted frequency, at every omitted cosine equals because all frequency ratios are odd. Thus the tail attains alternating extrema equal to its coefficient sum at points. The trigonometric Chebyshev alternation theorem proves the formula. For , , the largest decay exponent is .
A standard Weierstrass function is a uniformly convergent lacunary cosine series, here parameterized by and an integer frequency multiplier . Its continuity follows from the Weierstrass M-test. In the range , its modulus of continuity has the fractional-power scale for the positive odd-frequency model; the critical case can have the larger scale . These examples distinguish small approximation error from ordinary Lipschitz continuity.
For , all frequencies satisfy . At , the increment is . If , the first terms have and contribute at least , while the remaining tail has magnitude at most . This proves a lower modulus of continuity bound of order for small . The inverse theorem for trigonometric approximation supplies the matching upper bound because the positive lacunary trigonometric series has .
For positive summable coefficients, the Weierstrass M-test gives a continuous uniform sum. The partial sum over is the unique best uniform approximation of degree at most . If is the first omitted frequency, every tail term equals at because its frequency ratio is odd. The entire tail therefore attains alternating extrema equal in magnitude to its coefficient sum at more than points. The Chebyshev alternation theorem proves the assertion, including the empty partial sum at .
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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.