Handy Polynomial Fitting with Bernstein Polynomials

[Thomas Klemm]

(11-12-2018 01:21 PM)Thomas Okken Wrote:  I vaguely remember learning about Chebyshev polynomials for this purpose.

They are mentioned in the section: Change of interpolation points.

Quote: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...

It appears to be even better:

Quote:Therefore, when the interpolation nodes x_i are the roots of T_n, the error satisfies:
\(\left|f(x)-P_{{n-1}}(x)\right|\leq {\frac {1}{2^{{n-1}}n!}}\max _{{\xi \in [-1,1]}}\left|f^{{(n)}}(\xi )\right|\)

Cheers
Thomas