The main problem in using the inverse Abel transform of an experimentally observed function is that it involves numerical differentiation, which amplifies the errors affecting the measurements. This difficulty can be overcome by using spectral expansion methods, with coefficients obtainable through a numerical integration. The inaccuracy of the measurements is translated to these coefficients and is later propagated, with no amplification, to the inverse transform. Here, the authors propose a mixed analytical-numerical method for which the spectral expansion is chosen in terms of well known orthogonal functions, with coefficients numerically computed according to the precision of the data. They develop explicit expressions for the coefficients in the case of a data function in histogramatic form. The recurrence relations satisfied by the base functions facilitate the calculation of the coefficients. Therefore, the observed data can be represented, to an arbitrary degree of precision, with a small computational effort. In view of different possible applications, the authors consider in this paper two kinds of systems with finite or infinite dimensions.