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.

翻译：指数数据拟合的非线性逆问题是可分的，因为拟合函数是指数函数的线性组合，从而允许将线性系数与非线性参数分开求解。矩阵束方法将问题表述重新转化为非线性参数的广义特征值问题和线性参数的结构化线性系统，通常被视为求解该问题的更稳定计算方法。第2节总结了与经典复指数拟合或稀疏插值问题相关的矩阵束方法，并引入了膨胀和平移的概念以获得不同尺度下的矩阵束。指数分析早期已推广至多种多项式基函数和算子的本征函数。然而，在大多数推广中，缺乏基于特征值问题的计算方案。随后的第3至第6节将包括膨胀和平移范式的矩阵束方法推广至更多函数。每一类周期函数、多项式函数或特殊函数都需要定制化方法，并充分利用所考虑的稀疏插值问题中参数化初等函数或特殊函数的性质。每个推广都关联一个结构化线性矩阵束，从而直接通过广义特征值问题和一个或多个结构化线性系统分别得到非线性参数和线性参数的计算方案。最后，第7节展示了新方法的应用。