论文标题
Müntz定理中壮观的大型扩展系数
Spectacularly large expansion coefficients in Müntz's theorem
论文作者
论文摘要
例如，Müntz的定理断言，均匀的$ 1，x^2，x^4，\ dots $在$ c（[0,1]）$中密集。我们表明，相关的扩展效率低下，以至于与任何实际计算没有任何可能的相关性。例如，在此基础上，将$ f（x）= x $近似于准确性$ \ varepsilon = 10^{ - 6} $，需要大于$ x^{280 {，} 000} $的功率，并且系数大于$ 10^{107 {107 {，} 000} $。我们提出了一个定理，以建立相对于$ 1/\ varepsilon $的系数的指数增长。
Müntz'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 $1/\varepsilon$.
