I want to find a cubic polynomial $p$ approximating a continuous function $f$ over $[a,b]$. A critical property of $f$ is that $f(a)=a$ and $f(b)=b$, so $p$ must also satisfy $p(a)=a=f(a)$ and $ p(b)=b=f(b)$ exactly. Otherwise, $|f(x)-p(x)|$ should be small for each $x\in(a,b)$. This yields $2$ exact equations and $2$ remaining degrees of freedom in the coefficients of $p$. Some options I've tried for the remaining two coefficients include the following.
- Taking $p$ to be a cubic spline with $m_1=f'(a)$ and $m_2=f'(b)$ (I've found that this is not optimal)
- Interpolate $f$ on any roots of $f'$ (seems to yield better results, but can be over/underdetermined if $f'$ has too many/few roots in $[a,b]$)
Is there any theory of polynomial approximation that discusses a problem like this? Is it possible to project $f$ onto a set of orthogonal cubic polynomials such that $p(a)=a$ and $p(b)=b$? Would it be ideal to make $p-f$ equioscillate over $3$ nodes?
The optimal uniform approximation is indeed equi-oscillatory, but it’s hard to find. But the theory says that the interpolant at Chebyshev nodes is almost as good, and is trivially simple to obtain.
You worry that the Chebyshev interpolant only depends on four values of your function, and therefore might not capture its behaviour well. Maybe so, but if you want a better approximation, you’ll have to use a higher degree polynomial.
Another approach is to sample a large number of points from your function, and then fit a cubic polynomial to these points using a least-squares process. This might make you happier because you’re using more info from your function, but it won’t give you a better approximation.
I recommend you read Trefethen’s book “Approximation Theory and Practice” where all of this is nicely explained. A few chapters are available free on-line. Also Trefethen’s group has developed a Matlab add-in called Chebfun that provides a very convenient way to experiment with various approximation techniques.