Handy Polynomial Fitting with Bernstein Polynomials

[Thomas Okken]

(11-12-2018 05:39 AM)Thomas Klemm Wrote:  

(11-10-2018 10:22 PM)Namir Wrote:  I stumbled on an article discussing the advantages of using Bernstein polynomials for curve fitting. Unlike regular polynomials, the Bernstein polynomials offer smooth fitting with no wild deviations that occur when the order of the fitting classical polynomial is high.

This happens e.g. with Lagrange interpolation and is called Runge's phenomenon.
To mitigate the problem both Least squares fitting and approximation by using Bernstein polynomials are mentioned.
However these methods are not the same.

I vaguely remember learning about Chebyshev polynomials for this purpose. As I recall, Chebyshev fits have the nice property of having a hard upper bound on the error, which is within a constant (a factor of about 3 IIRC) of the worst-case error of the optimal fit. I'd have to dig around to find that textbook, though, it may have been lost in the mists of time...