Finding the min-max, or best , polynomial approximation to a function, in some standard interval, is of the greatest interest in numerical analysis   . For a polynomial function the least error distribution is a Chebyshev polynomial    .
The usual procedure   to find the best approximation to a general function is to start with a good approximation, say in the sense, easily obtained by the minimization of a quadratic functional for the coefficients, then iteratively improving this initial approximation by a Remez-like correction procedure   that strives to produce an error distribution that oscillates with a constant amplitude in the interval of interest.
In this note, we bring ample and varied computational evidence in support of the novel, worthy of notice, empirical numerical observation that taking the error distribution of a least squares, , best polynomial fit to a function, squared, as weight in a second, weighted, least squares approximation, results in an error distribution that is remarkably close to the best , or uniform, approximation.
2. Fixing Ideas; The Best Quadratic in [−1, 1]
The monic Chebyshev polynomial
is the solution of the min-max problem
This min-max solution, the least function in the sense, is a polynomial that has two distinct roots, and oscillates with a constant amplitude in Indeed, say is such a polynomial, and is another quadratic polynomial, then in the interval, for otherwise and would intersect at two points, which is absurd; is either an identity, or has but the one solution .
Thus, the monic Chebyshev polynomial of degree n is the least, uniform, or pointwise, error distribution in approximating by a polynomial of degree .
To obtain a least squares, a best , approximation to we first minimize
to have the value .
Minimizing next , under the weight
now with respect to p, we obtain , which is surprisingly much closer to the optimal value of one half.
We may replace the difficult measure by the computationally easier measure with an even . Let a0 be a good approximation, and be an improved one. Minimization cum linearization produces the equation
where is odd.
Starting with , we obtain from the above equation, for , the value , as compared with the optimal .
3. Optimal Cubic in [−1, 1]
Seeking to reproduce the optimal monic Chebyshev polynomial of degree three
we start by minimizing
and have .
Then we return to minimize the weighted with respect to
and obtain , which is considerably closer to the optimal value of 0.75. See Figure 1.
We are ready now for a Remez-like correction to bring the error function closer to optimal. The minimum of occurs at m = 0.50687. We write a new tentative and request that , by which we have
as compared with the Chebyshev optimal value of .
4. Optimal Quartic in [0, 1]
and obtain the best, in the sense, shown in Figure 2.
Then we return to minimize
weighted by the previous squared, and obtain the new, nearly perfectly uniform of Figure 3.
By comparison, the amplitude of the monic Chebyshev polynomial of degree four in [0,1] is 1/128 = 0.0078125.
Figure 1. (a) Least squares cubic. (b) Weighted least squares cubic.
Figure 2. Least squares quartic.
Figure 3. Weighted least squares quartic.
5. Best Cubic Approximation of ex in [0, 1]
To facilitate the integrations we use the approximation
with respect to . The best obtained from this minimization is shown in Figure 4.
Then we use the square of the minimal just obtained, as weight in the next minimization of
with respect to .
The nearly perfect result of this last minimization is shown in Figure 5.
6. Best Cubic Approximation of sinx in [0, 1]
To facilitate the integrations we take
and obtain the least squares error distribution as in Figure 6.
The subsequent nearly perfect weighted least squares error distribution is shown in Figure 7.
Figure 4. Least squares cubic fit to ex.
Figure 5. Weighted least squares cubic fit to ex.
Figure 6. Least squares cubic fit to sinx.
Figure 7. Weighted least squares cubic fit to sinx.
7. Best Quadratic Fit to in [0, 1]
We start with
under the condition
with respect to and , to have
shown as curve a in Figure 8.
Next we minimize
Figure 8. (a) Least squares quadratic fit to . (b) Weighted least squares quadratic fit to .
shown as graph b in Figure 8, as compared with the optimal, in the sense
8. Best Cubic Fit to x1/4 in [0, 1]
We start with
under the restriction , or , and minimize
with respect to to have the minimal shown in Figure 9.
Then we minimize
and obtain the nearly optimal error distribution as in Figure 10.
9. Another Difficult Function
We now look at the error distribution
under the condition that , or
Least squares minimization of yields the error distribution in Figure 11.
Next we minimize
Figure 9. Least squares cubic fit to .
Figure 10. Weighted least squares cubic fit to .
Figure 11. Least squares cubic fit to .
Figure 12. Weighted least squares cubic fit to .
under the restriction that , and obtain the nearly perfect error distribution shown in Figure 12.
We experimentally demonstrate, on a variety of continuous, analytic and nonanalytic functions, the remarkable observation that if the least squares polynomial approximation is taken as weight in a repeated, now weighted, least squares approximation, then this new, second, approximation is nearly perfect in the sense of Chebyshev, barely needing any further correction procedure.