A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube
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Numerical differentiation of a function, contaminated with noise, over the unit interval [0,1] ⊂ℝ by inverting the simple integration operator J:L^2([0,1]) → L^2([0,1]) defined as [Jx](s):=∫_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 σ_n(J) asymptotically proportional to 1/n, which indicates a degree 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) → L^2([0,1]^d) defined as [J_d x](s_1,…,s_d):=∫_0^s_1…∫_0^s_d x(t_1,…,t_d) dt_d… dt_1 over unit d-cube is to be inverted. This inverse problem of mixed differentiation x(s_1,…,s_d)=∂^d/∂ s_1 …∂ s_d y(s_1,… ,s_d) is of practical interest, for example when in statistics copula densities have to be verified from empirical copulas over [0,1]^d ⊂ℝ^d. In this note, we prove that the non-increasingly ordered singular values σ_n(J_d) of the operator J_d have an asymptotics of the form (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 ℛ(J_2) of the operator J_2.
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