  This paper intends to lay the theoretical foundation for the method of
functional maps, first presented in 2012 by Ovsjanikov, Ben-Chen, Solomon,
Butscher and Guibas in the field of the theory and numerics of maps between
shapes. We show how to analyze this method by looking at it as an application
of the theories of composition operators, of matrix representa- tion of
operators on separable Hilbert spaces, and of the theory of the Finite Section
Method. These are three well known fruitful topics in functional analysis. When
applied to the task of modelling of correspondences of shapes in
three-dimensional space, these concepts lead directly to functional maps and
its associated functional matrices. Mathematically spoken, functional maps are
composition operators between two-dimensional manifolds, and functional
matrices are infinite matrix representations of such maps. We present an
introduction into the notion and theoretical foundation of the functional
analytic framework of the theory of matrix repre- sentation, especially of
composition operators. We will also discuss two numerical methods for solving
equations with such operators, namely, two variants of the Rectangular Finite
Section Method. While one of these, which is well known, leads to an
overdetermined system of linear equations, in the second one the minimum-norm
solution of an underdetermined system has to be computed. We will present the
main convergence results related to these methods.
