Spectacularly large expansion coefficients in Müntz's theorem

Abstract

Müntz's theorem asserts, for example, that the even powers $1,x^2,x^4,\dots$ are dense in $C([0,1])$. We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating $f(x)=x$ to accuracy $\epsilon =10^{-6}$ in this basis requires powers larger than $x^{280,000}$ and coefficients larger than $10^{107,000}$. We present a theorem establishing exponential growth of coefficients with respect to $1/\epsilon$.