  The nonlinear inverse problem of exponential data fitting is separable since
the fitting function is a linear combination of parameterized exponential
functions, thus allowing to solve for the linear coefficients separately from
the nonlinear parameters. The matrix pencil method, which reformulates the
problem statement into a generalized eigenvalue problem for the nonlinear
parameters and a structured linear system for the linear parameters, is
generally considered as the more stable method to solve the problem
computationally. In Section 2 the matrix pencil associated with the classical
complex exponential fitting or sparse interpolation problem is summarized and
the concepts of dilation and translation are introduced to obtain matrix
pencils at different scales.
  Exponential analysis was earlier generalized to the use of several polynomial
basis functions and some operator eigenfunctions. However, in most
generalizations a computational scheme in terms of an eigenvalue problem is
lacking. In the subsequent Sections 3--6 the matrix pencil formulation,
including the dilation and translation paradigm, is generalized to more
functions. Each of these periodic, polynomial or special function classes needs
a tailored approach, where optimal use is made of the properties of the
parameterized elementary or special function used in the sparse interpolation
problem under consideration. With each generalization a structured linear
matrix pencil is associated, immediately leading to a computational scheme for
the nonlinear and linear parameters, respectively from a generalized eigenvalue
problem and one or more structured linear systems.
  Finally, in Section 7 we illustrate the new methods.
