The footprints of noise in rational interpolation

摘要

The reliability of rational approximations built from noisy data is explored. One considers a rational interpolation f(t) of a function S(t) known only via its, randomly perturbed, values at a finite number of real points. It is found that in the! small noise amplitude regime, the zeros and poles of f(t) split into two families: one reflecting the analytic structure of the function S(t); and one made of coalescent zero-pole pairs, most of the time confined in the real and/or complex vicinity of the interpolation interval. This exciting result seems to open the door to the detection of noise and, perhaps, to its subsequent erasure. [References: 10]