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

TL;DR

This paper demonstrates that M"untz's theorem expansions, while theoretically dense, are practically inefficient due to extremely large coefficients and high-degree terms needed for approximation.

Contribution

The paper proves that M"untz's theorem expansions have exponential coefficient growth, making them impractical for computational purposes.

Findings

Approximating simple functions requires extremely high-degree polynomials.

Coefficients in M"untz's expansions grow exponentially with inverse accuracy.

Practical use of these expansions is limited due to their inefficiency.

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 $1/\varepsilon$.