Application of the Padè method to solving the noisy trigonometric moment problem: some initial results

The noisy trigonometric moment problem for a finite linear combination of box functions is considered, and a research program, possibly leading to a superresolving method, is outlined and some initial steps are performed. The method is based on the remark that the poles of the Padè approximant to the Z-transform of the noiseless moments show, asymptotically, a regular pattern in the complex plane. The pattern can be described by a set of arcs, connecting points on the unit circle, and a pole density function defined on the arcs. When a moderate noise affects the moments, more arcs are needed to describe the pole pattern, but the noiseless pattern is slightly deformed, still allowing its identification. When this identification is possible, a very effective noise filter and moment extrapolator should be easily constructed. In this paper only some preliminary steps of the above research program are performed. Specifically, the case of one box function is considered. A method for computing the pole patterns, based on the solution of a singular integral equation of Cauchy type, is developed. The method is general enough to be used also for several box functions. Some numerical results, showing the feasibility of the program, are discussed.