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What do sparse interpolation, padé approximation, gaussian quadrature and tensor decomposition have in common?

We present the problem statement of sparse interpolation and its basic mathematical and computational methods to solve it. The solution is formulated already in 1795 by de Prony and only much later expressed in terms of a generalized eigenvalue problem and structured linear system. Input to the sparse interpolation methods are a very limited number of regularly collected samples. When considering these values as Taylor series coefficients, the problem statement easily connects to Padé approximation. The Padé approximant denominators are closely related to formal orthogonal polynomials and Gaussian quadrature. Results on their zeroes and certain convergence properties shed new light on some computational problems in sparse interpolation. The problem statement can also be viewed as a tensor decomposition problem, for which techniques from multilinear algebra can be used. Through the latter connection the toolkit of algorithms for sparse interpolation is further enlarged.