On data recovery with restraints on the spectrum range and the process range

TL;DR

This paper explores signal recovery from incomplete data by imposing spectral and process range restrictions, demonstrating conditions under which unique recovery is possible and robustness can be enhanced.

Contribution

It introduces a method to bypass the traditional uniqueness problem by restricting the process range and analyzing finite sequences with discretized processes.

Findings

Sequences can be dense in the space of all sequences under restrictions.

Uniqueness sets for these sequences can be singletons.

Additional observations improve robustness against data rounding.

Abstract

The paper considers recovery of signals from incomplete observations and a problem of determination of the allowed quantity of missed observations, i.e. the problem of determination of the size of the uniqueness sets for a given data recovery procedures. The paper suggests a way to bypass solution of this uniqueness problem via imposing restrictions investigates possibility of data recovery for classes of finite sequences under a special discretization of the process range. It is shown that these sequences can be dense in the space of all sequences and that the uniqueness sets for them can be singletons. Some robustness with respect to rounding of input data can be achieved via including additional observations.