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 $1/\varepsilon$.
翻译:M\"untz's the suorem 声称,比如说,偶数功率 1, x ⁇ 2, x ⁇ 4,\dts$ 以 $C ([0,0,1] 美元) 密度高。我们显示,相关的扩张效率低到无法想象任何实际计算。例如,以美元(x) =x美元接近x美元以精确度 $\varepsilon = 10 ⁇ 6}美元为基准,需要大于 $ 280{,}000} 美元和大于 10 ⁇ 107{,}000美元系数的功率。我们提出一个理论,确定1美元/\varepsilon美元系数的指数增长。