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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?

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I edited the formulas in your post using LaTeX. – user53153 Jan 28 '13 at 1:25
up vote 2 down vote accepted

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.

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