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Spectacularly large expansion coefficients in Müntz's theorem (2208.04176v1)

Published 8 Aug 2022 in math.NA and cs.NA

Abstract: M\"untz's theorem asserts, for example, that the even powers $1, x2, x4,\dots$ 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 $1/\varepsilon$.

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