Field reconstruction from measured values in symmetric volumes

A frequently encountered problem is that of determining the distribution of a physical quantity in a large, symmetric volume from a finite number of measured values. Below our investigation is based on invariance properties of the system equation from which the measured physical quantities are determined. No specific form of the system equation is used. We show that there are cases when the missing values can be found without information loss. We also give criteria to be satisfied by the measured values if this is the case, otherwise, the missing values will give rise to a larger variance indicating information loss. We point out that the perturbation caused by a disturbance possesses the same symmetry, as does the disturbance.