text: We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the roots of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony's approach to multivariate decomposition problems is presented , exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition , using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carath 'e odory-Fej 'e r decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.
  @cite_30: 
  @cite_4: What we know about multivariate Pade approximation has been developed in the last 25 years. In the next sections we compare and discuss many of these results. It will become clear that simple properties or requirements, such as the uniqueness of the Pade approximant and consequently its consistency property, can play a crucial role in the development of the multivariate theory. A separate section is devoted to a discussion of the convergence properties. At the end we include an extensive reference list on the topic.
  @cite_29: 
  @cite_52: Recently, a uniform approach was given by B. Beckermann and G. Labahn [Numer. Algorithms, 3 (1992), pp. 45-54] for different concepts of matrix-type Pade approximants, such as descriptions of vector and matrix Pade approximants along with generalizations of simultaneous and Hermite Pade approximants. The considerations in this paper are based on this generalized form of the classical scalar Hermite Pade approximation problem, power Hermite Pade approximation. In particular, this paper studies the problem of computing these new approximants. A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants. This recurrence also provides bases for particular subproblems. This generalizes previous work by Van Barel and Bultheel and, in a more general form, by Beckermann. The computation of the bases has complexity @math , where @math is the order of the desired approximant and requires no conditions on the input data. A second algorithm using the same recurrence relation along with divide-and-conquer methods is also presented. When the coefficient field allows for fast polynomial multiplication, this second algorithm computes a basis in the superfast complexity @math . In both cases the algorithms are reliable in exact arithmetic. That is, they never break down, and the complexity depends neither on any normality assumptions nor on the singular structure of the corresponding solution table. As a further application, these methods result in fast (and superfast) reliable algorithms for the inversion of striped Hankel, layered Hankel, and (rectangular) block-Hankel matrices.
  @cite_47: Abstract We obtain a several variables generalization of Kronecker's well-known result on finite rank Hankel matrices.
  @cite_17: The generalized eigenvalue problem @math with H a Hankel matrix and @math the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Pade approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled.
summary: Hankel matrices are central in the theory of Pad 'e approximants for functions of one variable. Here also a large literature exists for univariate Pad 'e approximants: see e.g. @cite_29 for approximation properties, @cite_17 for numerical stability problems, @cite_52 , @cite_30 for algorithmic aspects. The extension to multivariate functions is much less developed @cite_47 , @cite_4 .