The Journal of Online Mathematics and Its Applications, Volume 6 (2006)
The Chebyshev Equioscillation Theorem, Robert Mayans
Let f be a continuous function on the interval [a, b]. We have previously defined:
dn = inf{||f − p||: p is a polynomial on [a, b] of degree ≤ n}
Our goal is to show that this minimum is actually achieved: there exists a polynomial pn for which ||f − pn|| = dn. Such a polynomial pn is called a polynomial of best approximation of degree n to f. Our proof follows the argument in Isaacson and Keller (1994). Later we will show that for every f and n, the polynomial of best approximation is unique.
Let f, g be continuous real-valued functions on the interval [a, b]. Then
The proof is easy. The lemma says that the supremum norm satisfies the requirements of a vector space norm of the space of all continuous real-valued functions on [a, b]. Thus, we can define a metric on this space: the distance between f and g is d(f, g) = ||f − g||.
Let a = (a0, a1, ..., an) be a vector in Rn + 1. We define the polynomial associated with a:
pa(x) = a0 + a1 x1 + ··· + an xn
and the ordinary Euclidean length of a:
The Euclidean length defines a metric on Rn + 1 where the distance between a and b is d(a, b) = |b − a|.
The function mapping each a in Rn + 1 to ||pa|| is continuous.
We will show this when the interval is [0, 1]. Given ε > 0, let δ = ε / (n + 1). Let a = (a0, ..., an) and b = (b0, ..., bn), and suppose that |b − a| ≤ δ. Then for every i = 0, ..., n,
|bi − ai| ≤ |b − a| < ε / (n + 1).
Hence for any
x 
[0, 1],
|pb(x) − pa(x)| ≤ |b0 − a0| + |b1-a1| · |x| + ··· + |bn − an| · |xn| < ε / (n + 1) + ··· + ε / (n + 1) = ε.
We now can state and prove the main result of this section:
Let f be a continuous function on [a, b]. Then for every n, there is a polynomial pn(x) of degree ≤ n such that:
||f − pn|| = dn = inf {||f − q|| : q is a polynomial of degree } ≤ n}
Let
S = {a Rn + 1 : |a| = 1}
and let
ε = inf{|| pa|| : a S}.
Note that ε is the minimum value of a continuous function on S, a closed bounded set in
Rn + 1.
Hence there is an
a S
such that
ε = ||pa(x)||.
Further, we must have
ε > 0,
for otherwise
pa = 0
and
a = 0, and so a
cannot be in S.
Next, observe that ||pa|| tends to infinity as |a| tends to infinity, for
||pa|| = |a| · || pa / |a| || ≥ |a| · ε
which grows arbitrarily large as |a| tends to infinity.
Now choose M so large that ||pa|| ≥ dn + 1 + ||f|| whenever |a| ≥ M. Since
||f − p|| ≥ ||p|| − ||f|| ≥ dn + 1 + ||f|| − ||f|| = dn + 1
it follows that a polynomial pc of best approximation, if it exists, must satisfy |c| ≤ M.
Therefore
dn = inf{||f − pa|| : a Rn + 1 } = inf{||f − pa|| : |a| ≤ M}
and so
dn
is the minimum of a continuous function defined on the closed ball
{a Rn + 1 : |a| ≤ M}.
By compactness, there is a vector c within the closed ball that achieves the minimum value. For that vector c, we have
dn = ||f − pc|| and thus
pc
is a polynomial of best approximation.