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

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 in L2 is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σn(J) asymptotically proportional to 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 analog 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 dimensions d ϵ ℕ. Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J2. © 2023 Walter de Gruyter GmbH, Berlin/Boston.