Numerical differentiation of a function, contaminated with noise, over the unit interval $[0,1] \subset \mathbb{R}$ by inverting the simple integration operator $J:L^2([0,1]) \to L^2([0,1])$ defined as $[Jx](s):=\int_0^s x(t) dt$ is discussed extensively in the literature. The complete singular system of the compact operator $J$ is explicitly given with singular values $\sigma_n(J)$ asymptotically proportional to $1/n$, which indicates a degree {\sl one} of ill-posedness for this inverse problem. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case with operator $J$, there is little material available about the analysis of the d-dimensional case, where the compact integral operator $J_d:L^2([0,1]^d) \to L^2([0,1]^d)$ defined as $[J_d\,x](s_1,\ldots,s_d):=\int_0^{s_1}\ldots\int_0^{s_d} x(t_1,\ldots,t_d)\, dt_d\ldots dt_1$ over unit $d$-cube is to be inverted. This inverse problem of mixed differentiation $x(s_1,\ldots,s_d)=\frac{\partial^d}{\partial s_1 \ldots \partial s_d} y(s_1,\ldots ,s_d)$ is of practical interest, for example when in statistics copula densities have to be verified from empirical copulas over $[0,1]^d \subset \mathbb{R}^d$. In this note, we prove that the non-increasingly ordered singular values $\sigma_n(J_d)$ of the operator $J_d$ have an asymptotics of the form $\frac{(\log n)^{d-1}}{n}$, which shows that the degree of ill-posedness stays at one, even though an additional logarithmic factor occurs. Some more discussion refers to the special case $d=2$ for characterizing the range $\mathcal{R}(J_2)$ of the operator $J_2$.

翻译：本文广泛讨论了通过反演简单积分算子$J:L^2([0,1]) \to L^2([0,1])$（定义为$[Jx](s):=\int_0^s x(t) dt$）对单位区间$[0,1] \subset \mathbb{R}$上含有噪声的函数进行数值微分的问题。紧算子$J$的完整奇异系统被显式给出，其奇异值$\sigma_n(J)$渐近正比于$1/n$，表明该反问题的不适定度为{\sl 一}。本文回顾了Hilbert空间中具有紧正算子的线性算子方程的不适定度概念。与一维算子$J$的情形相比，关于d维情形（需反演定义在单位d维立方体上的紧积分算子$J_d:L^2([0,1]^d) \to L^2([0,1]^d)$，定义为$[J_d\,x](s_1,\ldots,s_d):=\int_0^{s_1}\ldots\int_0^{s_d} x(t_1,\ldots,t_d)\, dt_d\ldots dt_1$）的分析鲜有文献涉及。这一混合微分$x(s_1,\ldots,s_d)=\frac{\partial^d}{\partial s_1 \ldots \partial s_d} y(s_1,\ldots ,s_d)$的反问题具有实际意义，例如统计中需从$\mathbb{R}^d$中$[0,1]^d$上的经验copula验证copula密度时。本文证明算子$J_d$的非增序奇异值$\sigma_n(J_d)$具有$\frac{(\log n)^{d-1}}{n}$形式的渐近性，表明尽管出现额外对数因子，不适定度仍保持为一。关于算子$J_2$的值域$\mathcal{R}(J_2)$的刻画，进一步讨论了$d=2$的特殊情形。