Approximate N order polynomial as a weighted sum of lower order polynomials

Question

I want to represent a polynomial such as $x^5$ with a sum of weighted polynomials so that

$$x^5 - (ax^4 + bx^3 + cx^2 + dx + e) = \epsilon$$

My aim is to pick these weights $(a,b,c,d)$ assuming that the function is somewhere near the origin.

I recall seeing something like this before, what is it called?

Answer 1

The monic polynomial of smallest supremum norm on a given interval is a Chebyshev polynomial of first kind, with appropriate normalization. In particular, if your interval is $[-1,1]$, then $$2^{-4}T_5(x)=x^5 - \frac{5}{4}x^3+\frac{5}{16}x $$ does not exceed $1/16$ in absolute value, and this is the smallest possible supremum. Here is its graph:

and this is the comparison of $x^5$ and $(5/4)x^3-(5/16)x$:

If your interval is different, use a linear substitution and renormalize.