Related papers: Learning algebraic decompositions using Prony stru…
Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this note, we give a generalization of this…
We continue studying robustness of solving algebraic systems of Prony type (also known as the exponential fitting systems), which appear prominently in many areas of mathematics, in particular modern "sub-Nyquist" sampling theories. We show…
The paper gives an extension of Prony's method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H--bases.
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…
The paper considers a symbolic approach to Prony's method in several variables and its close connection to multivariate polynomial interpolation. Based on the concept of universal interpolation that can be seen as a weak generalization of…
We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the…
Eigenvalue analysis based methods are well suited for the reconstruction of finitely supported measures from their moments up to a certain degree. We give a precise description when Prony's method succeeds in terms of an interpolation…
We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian…
The generalized operator-based Prony method is an important tool for describing signals which can be written as finite linear combinations of eigenfunctions of certain linear operators. On the other hand, Bernoulli's algorithm and its…
Systems of Prony type appear in various signal reconstruction problems such as finite rate of innovation, superresolution and Fourier inversion of piecewise smooth functions. We propose a novel approach for solving Prony-type systems, which…
We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
We compare two known methods of extending a complex, unital, commutative normed algebra so as to include solutions to sets of monic polynomials over the original algebra. (One of these is a generalisation of a construction from the thesis…
In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…
In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…