In the previous post I mentioned that Remez algorithm computes the best polynomial approximation to a given function *f* as measured by the maximum norm. That is, for a given *n*, it finds the polynomial *p* of order *n* that minimizes the absolute error

|| *f* − *p* ||_{∞}.

The Mathematica function `MiniMaxApproximation`

minimizes the *relative *error by minimizing

|| (*f* − *p*) / *f* ||_{∞}.

As was pointed out in the comments to the previous post, Chebyshev approximation produces a nearly optimal approximation, coming close to minimizing the absolute error. The Chebyshev approximation can be computed more easily and the results are easier to understand.

To form a Chebyshev approximation, we expand a function in a series of Chebyshev polynomials, analogous to expanding a function in a Fourier series, and keep the first few terms. Like sines and cosines, Chebyshev polynomials are orthogonal functions, and so Chebyshev series are analogous to Fourier series. (If you find it puzzling to hear of functions being orthogonal to each other, see this post.)

Here is Mathematica code to find and plot the Chebyshev approximation to *e*^{x} over [−1, 1]. First, here are the coefficients.

weight[x_] := 2/(Pi Sqrt[1 - x^2]) c = Table[ Integrate[ Exp[x] ChebyshevT[n, x] weight[x], {x, -1, 1}], {n, 0, 5}]

The coefficients turn out to be exactly expressible in terms of Bessel functions, but typically you’d need to do a numerical integration with `NIntegrate`

.

Now we use the Chebyshev coefficients to evaluate the Chebyshev approximation.

p[x_] := Evaluate[c . Table[ChebyshevT[n - 1, x], {n, Length[c]}]] - First[c]/2

You could see the approximating polynomial with

Simplify[N[p[x]]]

which displays

1.00004 + 1.00002 x + 0.499197 x^2 + 0.166489 x^3 + 0.0437939 x^4 + 0.00868682 x^5

The code

Plot[Exp[x] - p[x], {x, -1, 1}]

shows the error in approximating the exponential function with the polynomial above.

Note that the plot has nearly equal ripples; the optimal approximation would have exactly equal ripples. The Chebyshev approximation is not optimal, but it is close. The absolute error is smaller than that of `MiniMaxApproximation`

by a factor of about *e*.

There are bounds on how different the error can be between the best polynomial approximation and the Chebyshev series approximation. For polynomials of degree *n*, the Chebyshev error is never more than

4 + 4 log(*n* + 1)/π

times the optimal approximation error. See Theorem 16.1 in Approximation Theory and Approximation Practice by Lloyd N. Trefethen.

An interesting property of the error curve is that it behaves like a concertina. That is, all the zeros and extreme points of the error curve are _monotone_ functions of one endpoint. You can see this numerically by shifting the interval [-1,1} to, say, [-0.9, 1] and then shifting the interval to [-0.9, 0,85]. In fact the entire curve shifts, hence the concertina. It might make a nice movie! Another property is that the approximating polynomial p_n(x) grows in magnitude more more rapidly _outside_ the interval of approximation than does any other polynomial of degree n with the same or smaller L-\infty error. Same extreme growth property holds for all the derivatives of p(x) too.