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

Abstract

Abstract 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 L 2 {L^{2}} is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σ n ⁢ ( J ) {\sigma_{n}(J)} asymptotically proportional to 1 n {\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 analog J d {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 ∈ ℕ {d\in{\mathbb{N}}} . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J 2 {J_{2}} .