Spectacularly large expansion coefficients in Müntz's theorem

08/08/2022

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by   Lloyd N. Trefethen, et al.

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Müntz's theorem asserts, for example, that the even powers 1, x^2, x^4,… 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 ε = 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/ε.