Over recent years; tracing back sources of chemical species dispersed through the atmosphere has been of considerable importance, with an emphasis on increasing the precision of the source resolution. This need stems from many problems: being able to estimate the emissions of pollutants; spotting the source of radionuclides; evaluating diffuse gas fluxes; etc. We study the high-resolution retrieval on a continental scale of the source of a passive atmospheric tracer, given a set of concentration measurements. In the first of this two-part paper, we lay out and develop theoretical grounds for the reconstruction. Our approach is based on the principle of maximum entropy on the mean. It offers a general framework in which the information input prior to the inversion is used in a flexible and controlled way. The inversion is shown to be equivalent to the minimization of an optimal cost function, expressed in the dual space of observations. Examples of such cost functions are given for different priors of interest to the retrieval of an atmospheric tracer. In this respect, variational assimilation (4D-Var), as well as projection techniques, are obtained as biproducts of the method. The framework is enlarged to incorporate noisy data in the inversion scheme.