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
`p`_{n}(`x`)
is a polynomial of best approximation of degree `n` if and only if
`f`, `p`_{n}
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
`d`_{n} > 0.

Suppose that
`f`, `p`_{n}
has an alternating set of length
`n` + 2.
By Theorem 4, we have
|| `f` − `p`_{n} || ≤ `d`_{n}.
As
`d`_{n} ≤ || `f` − `p`_{n} ||
by the definition of
`d`_{n},
it follows that
`p`_{n}
is a polynomial of best approximation to `f`.

Now suppose that
`p`_{n}
is a polynomial of best approximation to `f`. By Lemma 4,
`f`, `p`_{n}
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
`p`_{n}
and get a better approximation than
`p`_{n},
which is impossible. Thus every polynomial of best approximation has an alternating set of length at least `n` + 2.

To show uniqueness, suppose that
`p`_{n}
and
`q`_{n}
are both polynomials of best approximation, and we will show that they are equal.

Note that
(`p`_{n} + `q`_{n}) / 2
is a polynomial of best approximation, as:

Therefore, there are
`n` + 2
alternating points at which
(`f` − `p`_{n}) / 2 + (`f` − `q`_{n}) / 2 = ± `d`_{n}.

At each of these alternating points,
`f` − `p`_{n}
and
`f` − `q`_{n}
are both
`d`_{n}
or both
−`d`_{n}.
So
`f` − `p`_{n}
and
`f` − `q`_{n}
agree on
`n` + 2
points, and so
(`f` − `p`_{n}) − (`f` − `q`_{n}) = `q`_{n} − `p`_{n} = 0
at these
`n` + 2
points. Since
`q`_{n} − `p`_{n}
is a polynomial of degree
≤ `n`,
`q`_{n}
and
`p`_{n}
must be identical. Therefore the polynomial
`p`_{n}
of best approximation is unique.

The polynomial
`p`_{n}
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
2`n` + 1
when
`p`(`x`) = 0.
By the Chebyshev theorem, the polynomial
`p`_{m}
of best approximation for
`m` ≤ 2`n` − 1
is
`p`_{m}(`x`) = 0.