Spectacularly Large Expansion Coefficients in Müntz’s Theorem

Abstract

AbstractMüntz’s theorem asserts, for example, that the linear span of the even powers$$1, x^2, x^4,\dots $$1,x2,x4,⋯is dense in$$C([0,1])$$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$$f(x)=x$$f(x)=xto accuracy$$\varepsilon = 10^{-6}$$ε=10-6in this basis requires powers larger than$$x^{280{,}000}$$x280,000and coefficients larger than$$10^{107{,}000}$$10107,000. We present a theorem establishing exponential growth of coefficients with respect to$$1/\varepsilon $$1/ε.