The Journal of Online Mathematics and Its Applications, Volume 6 (2006)

The Chebyshev Equioscillation Theorem, Robert Mayans

Given a continuous function `f` on a closed interval
[`a`, `b`],
how closely can we approximate `f` by a polynomial on
[`a`, `b`]?
This is a central question in the theory of approximation of functions. To sharpen the question, let us define the supremum norm of `f` to be:

and the best minimax error of degree `n` by

`d`_{n} = inf{||`f` − `p`||: `p` is a polynomial on [`a`, `b`] of degree ≤ `n`}

The Weierstrass Approximation Theorem says that there is a sequence of polynomials that converge uniformly to `f`, so that
`d`_{n} → 0
as
`n` → 0.
We are interested in finding, for any degree `n`, the polynomial
`p`_{n},
called the polynomial of best approximation, that minimizes
|`f` − `p`_{n}|.

The Chebyshev Equioscillation Theorem asserts that the polynomial of best approximation has the following striking pattern: it overestimates and underestimates `f` by exactly
`d`_{n}
in an alternating pattern, at least
`n` + 2
times, and it is the unique polynomial of degree
≤ `n`
to do so.

The following figure shows a plot of
sin^{2}(`x`) from
−`π` / 2
to
`π` / 2,
and the quadratic polynomial
`p`(`x`)
of best approximation:

The polynomial
`p`(`x`)
overestimates the function at the endpoints
`a` = `π` / 2,
−`a` = −`π` / 2
and at
`x` = 0.
It most underestimates the function at the points
± `z`,
where
`z` = 1.056612311.
The errors are equal and maximal at these five points. We call the arrangement of points
(−`π` / 2, −`z`, 0, `z`, `π` / 2)
an alternating set of length 5.

The Chebyshev theorem is a result of great influence in the theory of polynomial approximation. But the theorem and its proof are usually omitted from the undergraduate numerical analysis course. To quote one text [Atkinson (1989)]: "The proof is quite technical, amounting to a complicated and manipulatory proof by contradiction." Our aim in this paper is to use applets to motivate and illustrate the proof of this theorem. The proof should be accessible to students with a basic course in real analysis, covering metric spaces, uniform continuity, compactness of closed and bounded sets in
`R`^{n},
and so on. Even for students with less preparation, the main idea of the proof can be grasped clearly: if a polynomial approximation of degree `n` does not produce an alternating set of length
`n` + 2,
then we have an intuitive procedure, described by the applets, for improving the polynomial approximation.