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Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator  J  mapping inL2{L^{2}}is discussed extensively in the literature. The complete singular system of the compact operator  J  is explicitly given with singular valuesσn⁢(J){\sigma_{n}(J)}asymptotically proportional to1n{\frac{1}{n}}. This indicates a degree  one  of ill-posedness for the associated inverse problem of differentiation. 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, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analogJd{J_{d}}to  J , defined over unit  d -cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensionsd∈ℕ{d\in{\mathbb{N}}}. Some more discussion refers to the two-dimensional case in order to characterize the range of the operatorJ2{J_{2}}.

A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube