Analytic continuation, singular-value expansions, and Kramers-Kronig analysis

We describe a systematic approach to the recovery of a function analytic in the upper half-plane, C+, from measurements over a finite interval on the real axis, D subset of R. Analytic continuation problems of this type are well known to be ill-posed. Thus, the best one can hope for is a simple, linear procedure which exposes this underlying difficulty and solves the problem in a least-squares sense. To accomplish this, we first construct an explicit analytic approximation of the desired function and recast the continuation problem in terms of a 'residual function' defined on the measurement window D itself. The resulting procedure is robust in the presence of noise, and we demonstrate its performance with some numerical experiments.