ILL-Posedness and Precision in Object Field Reconstruction Problems

We suppose σ(x), called the object field, is an unknown real or complex-valued function on a set E. Our knowledge of σ is given by observation, with some (small) error, of a real or complex-valued function s(ξ) given by s(ξ)=[Bσ](ξ)=∫ E b(ξ,x)σ(x)dx,     σ ε F The sets E,F are subsets of Rn, where normally n=1,2,3. It is desired to determine σ(x) as well as possible; the result of this determination is denoted σ^(x) (in ideal circumstances σ^(x)=σ(x)) and is called an estimate of σ (the word estimate may or may not have a statistical implication). An inversion problem is determined when the following are specified: (1) the kernel b(ξ,x) of the integral operator; (2) the region E on which σ(x) is defined; (3) the region F over which the observation is made; (4) the set ∑ of functions σ that are allowed. It is assumed that the situation is such that σ ε L2 (E) (the space of square-integrable functions on E) and s ε L2 (F). If ∑ is all of L2 (E), the problem is an unconstrained linear inversion; if ∑ is not all of L2 (E) (∑ may be linear or nonlinear) it is a constrained linear inversion.