The Journal of Online Mathematics and Its Applications, Volume 6 (2006)
The Chebyshev Equioscillation Theorem, Robert Mayans
Let f be a continuous real-valued function on [a, b]. Then pn(x) is a polynomial of best approximation of degree n if and only if f, pn has an alternating set of length n + 2. Furthermore, the polynomial of best approximation is unique.
The theorem is trivially true if f is itself a polynomial of degree ≤ n. We assume not, and so dn > 0.
Suppose that f, pn has an alternating set of length n + 2. By Theorem 4, we have || f − pn || ≤ dn. As dn ≤ || f − pn || by the definition of dn, it follows that pn is a polynomial of best approximation to f.
Now suppose that pn is a polynomial of best approximation to f. By Lemma 4, f, pn has an alternating set of length 2, and by Theorem 5, it can be extended into a sectioned alternating set of length m. We must have m ≥ n + 2, for if m ≤ n + 1 then by Lemma 6, we could add a polynomial q of degree ≤ n to pn and get a better approximation than pn, which is impossible. Thus every polynomial of best approximation has an alternating set of length at least n + 2.
To show uniqueness, suppose that pn and qn are both polynomials of best approximation, and we will show that they are equal.
Note that (pn + qn) / 2 is a polynomial of best approximation, as:
Therefore, there are n + 2 alternating points at which (f − pn) / 2 + (f − qn) / 2 = ± dn.
At each of these alternating points, f − pn and f − qn are both dn or both −dn. So f − pn and f − qn agree on n + 2 points, and so (f − pn) − (f − qn) = qn − pn = 0 at these n + 2 points. Since qn − pn is a polynomial of degree ≤ n, qn and pn must be identical. Therefore the polynomial pn of best approximation is unique.
The polynomial pn of best approximation may have degree < n, and may produce an alternating set of length > n + 2. As an example of both, consider the polynomial of best approximation of degree m to the function y = f(x) = cos(x) on the interval [−n / π, n / π]. There is an obvious alternating set of length 2n + 1 when p(x) = 0. By the Chebyshev theorem, the polynomial pm of best approximation for m ≤ 2n − 1 is pm(x) = 0.