论文标题
关于Hausdorff时刻问题的分析和数字问题的开放问题
Open questions asked to analysis and numerics concerning the Hausdorff moment problem
论文作者
论文摘要
我们解决了有关复合hausdorff力矩问题的不良程度的事实和开放问题，该问题旨在从l^2（0,1）$中恢复函数$ x \ y^2（0,1）$从无限尺寸序列空间$ \ ell^2 $中的元素中，该元素表征了适用于$ x $ $ x $ $ x $的触发器的瞬间。到目前为止，该学位是由于相关紧凑型前向操作员$ a $的奇异值的衰减速率所产生的，这是紧凑型简单集成运算符映射在$ l^2（0,1）$中的组成，而非compact Hausdorff Moment Operator Moment Operator $ B^{（h）} $从$ l^2（0,11）$ $^2 to $^2 to $^2 to $ \^2 to $^2 to $。 There is a seeming contradiction between (a) numerical computations, which show (even for large $n$) an exponential decay of the singular values for $n$-dimensional matrices obtained by discretizing the operator $A$, and \linebreak (b) a strongly limited smoothness of the well-known kernel $k$ of the Hilbert-Schmidt operator $A^*A$.事实（a）表明，无限维度的豪斯多夫时刻问题的严重不良性，而事实（b）让我们期望相反，因为指数不良发生在共同的情况下，仅以$ c^\ infty $ kernels $ k $。我们回想起可能出现$ a $的单数值的多项式衰减的论点，即使数字似乎是反对它的，并讨论了非紧凑型操作员的数值近似中的某些问题。
We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function $x \in L^2(0,1)$ from elements of the infinite dimensional sequence space $\ell^2$ that characterize moments applied to the antiderivative of $x$. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator $A$, which is the composition of the compact simple integration operator mapping in $L^2(0,1)$ and the non-compact Hausdorff moment operator $B^{(H)}$ mapping from $L^2(0,1)$ to $\ell^2$. There is a seeming contradiction between (a) numerical computations, which show (even for large $n$) an exponential decay of the singular values for $n$-dimensional matrices obtained by discretizing the operator $A$, and \linebreak (b) a strongly limited smoothness of the well-known kernel $k$ of the Hilbert-Schmidt operator $A^*A$. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness occurs in common just for $C^\infty$-kernels $k$. We recall arguments for the possible occurrence of a polynomial decay of the singular values of $A$, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.
