Spectacularly large expansion coefficients in M\"untz's theorem

Abstract

M\"untz'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 $\varepsilon = 10^{-6}$ 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 $1/\varepsilon$.