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：关于在单位区间$[0,1] \subset \mathbb{R}$上对含噪声函数进行数值微分的问题，通过求逆简单积分算子$J:L^2([0,1]) \to L^2([0,1])$（定义为$[Jx](s):=\int_0^s x(t) dt$）已在文献中得到广泛讨论。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$1/n$，表明该反问题的病态程度为{\sl 一}。本文回顾了希尔伯特空间中紧正向算子线性算子方程的病态程度概念。与一维情形下的算子$J$相反，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$J_d:L^2([0,1]^d) \to L^2([0,1]^d)$，其定义为$[J_d\,x](s_1,\ldots,s_d):=\int_0^{s_1}\ldots\int_0^{s_d} x(t_1,\ldots,t_d)\, dt_d\ldots dt_1$）的分析资料很少。这种混合微分反问题$x(s_1,\ldots,s_d)=\frac{\partial^d}{\partial s_1 \ldots \partial s_d} y(s_1,\ldots ,s_d)$具有实际应用价值，例如在统计学中需通过$[0,1]^d \subset \mathbb{R}^d$上的经验copula验证copula密度时。本文证明了算子$J_d$的非递增序奇异值$\sigma_n(J_d)$具有渐近形式$\frac{(\log n)^{d-1}}{n}$，表明尽管出现额外对数因子，其病态程度仍保持为一。额外讨论涉及特殊情形$d=2$下算子$J_2$的值域$\mathcal{R}(J_2)$的刻画。