Polynomial–Exponential Decomposition From Moments

We analyze the decomposition problem of multivariate polynomial–exponential functions from their 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 algebr...

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Veröffentlicht in:	Foundations of computational mathematics Jg. 18; H. 6; S. 1435 - 1492
1. Verfasser:	Mourrain, Bernard
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2018 Springer Nature B.V Springer Verlag
Schlagworte:	Algebra Algebraic Geometry Algorithms Applications of Mathematics Computer Science Convolution Decomposition Economics Eigenvectors Exponential functions Hankel matrices Interpolation Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Operators (mathematics) Partial differential equations Polynomials Power series Theorems Polynomial–exponential series 68W30 Prony Differential equation 14Q20 15B05 Artinian Hankel matrix Interpolation Gorenstein Moment 47B35 Sparse representation AMS classification: 14Q20 15B05 1
ISSN:	1615-3375, 1615-3383
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Abstract	We analyze the decomposition problem of multivariate polynomial–exponential functions from their 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 isolated points 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 Carathéodory–Fejér 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.
AbstractList	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 Carathéodory-Fejér 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. We analyze the decomposition problem of multivariate polynomial–exponential functions from their 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 isolated points 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 Carathéodory–Fejér 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.
Author	Mourrain, Bernard
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Cites_doi	10.1088/0266-5611/19/2/201 10.1007/BF02948939 10.1137/S0895479892230031 10.1016/S0024-3795(98)10223-9 10.1016/j.laa.2010.06.046 10.1007/3-540-09519-5_73 10.1109/29.32276 10.1137/110836584 10.1023/A:1011901522520 10.1016/j.sigpro.2009.11.012 10.3792/chmm/1263317740 10.1145/2755996.2756673 10.1007/978-1-4757-2181-2 10.1088/0266-5611/29/2/025001 10.1007/978-0-387-09686-5_7 10.1016/j.laa.2015.10.023 10.2174/97816080504821100101 10.1109/TIT.1968.1054109 10.1017/CBO9781139856065 10.1016/j.acha.2009.09.001 10.1007/s00013-009-0007-6 10.1145/1073884.1073920 10.1016/S0747-7171(88)80033-6 10.1145/62212.62241 10.1007/BF03014797 10.5802/aif.65 10.1090/mmono/002 10.1109/78.143447 10.1145/1576702.1576727 10.1007/3-540-46796-3_41 10.1007/978-3-540-71647-1 10.1016/0024-3795(82)90110-0 10.1109/TIT.2013.2291876 10.1007/s00211-016-0844-8 10.1006/jcom.1999.0530 10.1007/BF01199078 10.1016/j.jsc.2008.11.003 10.1007/s00211-006-0054-x 10.1007/BF03014796 10.1016/j.jsc.2012.05.012 10.1007/s11785-016-0596-6 10.1006/aima.1998.1782 10.1002/gamm.201410011 10.1109/18.59953 10.1016/j.jsc.2012.03.007 10.1017/CBO9780511530074 10.1002/cpa.20124 10.1007/BFb0093426 10.1016/j.acha.2005.01.003 10.1007/s00020-015-2217-6 10.1016/S0377-0427(99)00028-X 10.1109/TIT.1969.1054260 10.1109/TIT.2016.2553041
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Keywords	Polynomial–exponential series 68W30 Prony Differential equation 14Q20 15B05 Artinian Hankel matrix Interpolation Gorenstein Moment 47B35 Sparse representation AMS classification: 14Q20 15B05 1
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References	Constantin Carathéodory and Lipòt Fejér. Über den Zusammenhang der Extremen von Harmonischen Funktionen mit Ihren Koeffizienten und Über den Picard-Landauschen Satz. In Rendiconti del Circolo Matematico di Palermo (1884-1940), volume 32, pages 218–239. 1911. Stef Graillat and Philippe Trébuchet. A new algorithm for computing certified numerical approximations of the roots of a zero-dimensional system. In Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pages 167–174. ACM, 2009. CaixingGuFinite rank Hankel operators on the polydiskLinear Algebra and its Applications1999288269281167057910.1016/S0024-3795(98)10223-9 BrachatJéromeComonPierreMourrainBernardTsigaridasEliasSymmetric tensor decompositionLinear Algebra and Applications201043318511872273610310.1016/j.laa.2010.06.046 BeckermannBernhardGolubGene HLabahnGeorgeOn the numerical condition of a generalized Hankel eigenvalue problemNumerische Mathematik200710614168228600610.1007/s00211-006-0054-x Bernard Mourrain and Philippe Trébuchet. Generalized normal forms and polynomials system solving. In M. Kauers, editor, Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’05), pages 253–260, 2005. PowerStephen CFinite rank multivariable Hankel formsLinear Algebra and its Applications19824823724468322110.1016/0024-3795(82)90110-0 BerlekampElwyn RNonbinary BCH decodingIEEE Transactions on Information Theory196814224224210.1109/TIT.1968.1054109 Riche de PronyBaron GaspardEssai expérimental et analytique: sur les lois de la dilatabilité de fluides élastiques et sur celles de la force expansive de la vapeur de l’alcool, à différentes températuresJ. École Polytechnique179512476 Francis S. Macaulay. The Algebraic Theory of Modular Systems. Cambridge University Press, 1916. PlonkaGerlindTascheManfredProny methods for recovery of structured functionsGAMM-Mitteilungen2014372239258328502210.1002/gamm.201410011 FischerErnstÜber das Carathéodory’sche Problem, Potenzreihen mit positivem reellen Teil betreffendRendiconti del Circolo Matematico di Palermo (1884–1940)191132124025610.1007/BF03014797 RiquierCharlesLes systèmes d’équations aux dérivées partielles1910ParisGauthier-Villars40.0411.01 LimLek-HengComonPierreBlind multilinear identificationIEEE Transactions on Information Theory201460212601280316497410.1109/TIT.2013.2291876 Bernard Mourrain. A new criterion for normal form algorithms. In M. Fossorier, H. Imai, Shu Lin, and A. Poli, editors, Proc. AAECC, volume 1719 of LNCS, pages 430–443. Springer, Berlin, 1999. FliessMichelSéries reconnaissables, rationnelles et algébriquesBulletin des Sciences Mathématiques. Deuxième Série1970942312390219.16003 N. E. Golyandina and K. D. Usevich. 2D-Extension of Singular Spectrum Analysis: Algorithm and Elements of Theory. In Matrix Methods: Theory, Algorithms and Applications, pages 449–473. World Scientific Publishing, 2010. LasserreJean-BernardLaurentMoniqueMourrainBernardRostalskiPhilippTrébuchetPhilippeMoment matrices, border bases and real radical computationJournal of Symbolic Computation2013516385300578210.1016/j.jsc.2012.03.007 Leopold Kronecker. Zur Theorie der Elimination Einer Variabeln aus Zwei Algebraischen Gleichungen. pages 535–600, 1880. MasseyJamesShift-register synthesis and BCH decodingIEEE transactions on Information Theory196915112212724255610.1109/TIT.1969.1054260 The MUSIC AlgorithmA. Lee Swindlehurst and Thomas Kailath. A Performance Analysis of Subspace-Based Methods in the Presence of Model Errors - Part IIEEE Trans. on Signal Processing1992401758177410.1109/78.143447 YangZaiXieLihuaStoicaPetreVandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-ResolutionIEEE Transactions on Information Theory201662636853701350675710.1109/TIT.2016.2553041 Encyclopedia of Mathematics, April 2016. AnderssonFredrikCarlssonMarcusOn General Domain Truncated Correlation and Convolution Operators with Finite RankIntegral Equations and Operator Theory2015823339370335578410.1007/s00020-015-2217-6 Victor Pereyra and Godela Scherer, editors. Exponential Data Fitting and Its Applications. Bentham Science Publisher, 2012. GolubGenePereyraVictorSeparable nonlinear least squares: The variable projection method and its applicationsInverse Problems2003192R1R26199178610.1088/0266-5611/19/2/201 AnderssonFredrikCarlssonMarcusde HoopMaarten VNonlinear approximation of functions in two dimensions by sums of wave packetsAppl. Comput. Harmon. Anal.2010292198213265245810.1016/j.acha.2009.09.001 MacWilliamsJessie FSloaneNeil J AThe Theory of Error-Correcting Codes, Volume 161977AmsterdamNorth Holland Publishing Co.0369.94008 George A. Baker and Peter Graves-Morris. Padé Approximants. Cambridge University Press, Cambridge England, New York, 2nd edition, 1996. CurtoRaul EFialkowLawrence ASolution of the Truncated Complex Moment Problem for Flat Data1996Providence, R.IAmer Mathematical Society0876.30033 SchwartzLaurentThéorie des distributions1966ParisEditions Hermann0149.09501 Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM ’79, pages 216–226, London, UK, 1979. Springer-Verlag. IarrobinoAnthonyKanevVassilPower Sums, Gorenstein Algebras, and Determinantal Loci1999BerlinSpringer10.1007/BFb0093426 Fredrik Andersson and Marcus Carlsson. On the structure of positive semi-definite finite rank general domain Hankel and Toeplitz operators in several variables. Complex Analysis and Operator Theory, 11(4):755–784, 2017. BernardiAlessandraBrachatJérômeComonPierreMourrainBernardGeneral tensor decomposition, moment matrices and applicationsJournal of Symbolic Computation2013525171301812810.1016/j.jsc.2012.05.012 LaurentMoniqueMourrainBernardA generalized flat extension theorem for moment matricesArchiv der Mathematik20099318798252064710.1007/s00013-009-0007-6 MourrainBernardIsolated points, duality and residuesJ. of Pure and Applied Algebra1996117&11846949314578510896.13020 MourrainBernardPanVictor YMultivariate Polynomials, Duality, and Structured MatricesJournal of Complexity2000161110180176240110.1006/jcom.1999.0530 BeckermannBernhardLabahnGeorgeA uniform approach for the fast computation of matrix-type Padé approximantsSIAM Journal on Matrix Analysis and Applications1994153804823128269610.1137/S0895479892230031 PeterThomasPlonkaGerlindA generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operatorsInverse Problems2013292025001301196410.1088/0266-5611/29/2/025001 CuytAnnieHow well can the concept of Padé approximant be generalized to the multivariate case?Journal of Computational and Applied Mathematics19991051–22550169057710.1016/S0377-0427(99)00028-X PedersenPaul SBasis for Power Series Solutions to Systems of Linear, Constant Coefficient Partial Differential EquationsAdvances in Mathematics19991411155166166714910.1006/aima.1998.1782 RochbergRichardToeplitz and Hankel operators on the Paley–Wiener spaceIntegral Equations and Operator Theory198710218723587824610.1007/BF01199078 RoyR.KailathT.ESPRIT-estimation of signal parameters via rotational invariance techniquesIEEE Transactions on Acoustics, Speech, and Signal Processing198937798499510.1109/29.32276 ElkadiMohamedMourrainBernardIntroduction à la résolution des systèmes polynomiaux, Mathématiques & Applications2007BerlinSpringer10.1007/978-3-540-71647-1 MalgrangeBernardExistence et approximation des solutions des équations aux dérivées partielles et des équations de convolutionAnnales de l’institut Fourier195662713558699010.5802/aif.65 OberstUlrichPauerFranzThe Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant CoefficientsMultidimensional Systems and Signal Processing2001123–4253308186881110.1023/A:1011901522520 Jérémy Berthomieu, Brice Boyer, and Jean-Charles Faugère. Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences. In International Symposium on Symolic and Algebraic Compution, pages 61–68. ACM Press, 2015. Michael Ben-Or and Prasson Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In Proceedings of the twentieth annual ACM symposium on Theory of computing, STOC ’88, pages 301–309, New York, NY, USA, 1988. ACM. BarachartLaurentSur la réalisation de Nerode des systèmes multi-indicielsC. R. Acad. Sc. Paris1984301715718 PottsDanielTascheManfredParameter estimation for multivariate exponential sumsElectronic Transactions on Numerical Analysis20134020422430815701305.65093 EmsalemJacquesGéométrie des points épaisBulletin de la S.M.F.19781063994160396.13017 HormanderLarsAn Introduction to Complex Analysis in Several Variables19903AmsterdamNorth Holland0685.32001 CandèsEmmanuel JRombergJustinTaoTerenceStable signal recovery from incomplete and inaccurate measurementsCommunications on Pure and Applied Mathematics200659812071223223084610.1002/cpa.20124 SauerTomasProny’s method in several variablesNumerische Mathematik20171362411438364809310.1007/s00211-016-0844-8 BeylkinGregoryMonzónLucasOn approximation of functions by exponential sumsAppl. Comput. Harmon. Anal.20051911748214706010.1016/j.acha.2005.01.003 SakataShojiroFinding a minimal set of linear recurring relations capable of generating a given finite two-dimensional arrayJournal of Symbolic Computation19885332133794658710.1016/S0747-7171(88)80033-6 James Joseph Sylvester. Essay on Canonical Form. The collected mathematical papers of J. J. Sylvester, Vol. I, Paper 34, Cambridge University Press. 1909 (XV und 688). G. Bell, London, 1851. Naum I. Ahiezer and Mark G. Kreĭn. Some Questions in the Theory of Moments. American Mathematical Society, Providence, 1962. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and Its Applications, volume 149, pages 157–270. Springer, New York, 2009. KunisStefanPeterThomasRömerTimvon der OheUlrichA mul Charles Riquier (9372_CR61) 1910 David A Cox (9372_CR19) 1992 9372_CR49 Michel Fliess (9372_CR28) 1970; 94 9372_CR44 Dmitry Batenkov (9372_CR8) 2013; 73 Ernst Fischer (9372_CR26) 1911; 32 9372_CR41 Bernard Mourrain (9372_CR50) 2000; 16 Patrick Fitzpatrick (9372_CR27) 1990; 36 Monique Laurent (9372_CR42) 2009; 93 9372_CR1 9372_CR2 9372_CR4 Bernard Malgrange (9372_CR46) 1956; 6 9372_CR6 Lek-Heng Lim (9372_CR43) 2014; 60 9372_CR38 Fredrik Andersson (9372_CR5) 2010; 29 9372_CR32 9372_CR33 Bernard Mourrain (9372_CR48) 1996; 117&118 Daniel Potts (9372_CR58) 2010; 90 9372_CR31 Laurent Schwartz (9372_CR66) 1966 Mark Giesbrecht (9372_CR29) 2009; 44 Anthony Iarrobino (9372_CR37) 1999 Elwyn R Berlekamp (9372_CR12) 1968; 14 Jacques Emsalem (9372_CR25) 1978; 106 9372_CR71 Ulrich Oberst (9372_CR52) 2001; 12 Gene Golub (9372_CR30) 2003; 19 Paul S Pedersen (9372_CR53) 1999; 141 Stefan Kunis (9372_CR39) 2016; 490 Laurent Barachart (9372_CR7) 1984; 301 Tomas Sauer (9372_CR65) 2017; 136 Emmanuel J Candès (9372_CR17) 2006; 59 Vladimir V Peller (9372_CR54) 1998; 33 Jérome Brachat (9372_CR16) 2010; 433 9372_CR68 9372_CR63 Bernhard Beckermann (9372_CR10) 1994; 15 Gregory Beylkin (9372_CR15) 2005; 19 Jean-Bernard Lasserre (9372_CR40) 2013; 51 Jessie F MacWilliams (9372_CR45) 1977 Shojiro Sakata (9372_CR64) 1988; 5 Stephen C Power (9372_CR60) 1982; 48 Richard Rochberg (9372_CR62) 1987; 10 Bernhard Beckermann (9372_CR9) 2007; 106 Zai Yang (9372_CR70) 2016; 62 9372_CR18 Annie Cuyt (9372_CR21) 1999; 105 Alessandra Bernardi (9372_CR13) 2013; 52 9372_CR14 Raul E Curto (9372_CR20) 1996 9372_CR11 David Eisunbud (9372_CR23) 1994 9372_CR55 Lars Hormander (9372_CR36) 1990 9372_CR51 Gerlind Plonka (9372_CR57) 2014; 37 Thomas Peter (9372_CR56) 2013; 29 Joachim von zur Gathen (9372_CR69) 2013 James Massey (9372_CR47) 1969; 15 Baron Gaspard Riche de Prony (9372_CR22) 1795; 1 Gu Caixing (9372_CR34) 1999; 288 Fredrik Andersson (9372_CR3) 2015; 82 Mohamed Elkadi (9372_CR24) 2007 Hakop A Hakopian (9372_CR35) 2004; 10 The MUSIC Algorithm (9372_CR67) 1992; 40 Daniel Potts (9372_CR59) 2013; 40
References_xml	– reference: FliessMichelSéries reconnaissables, rationnelles et algébriquesBulletin des Sciences Mathématiques. Deuxième Série1970942312390219.16003 – reference: PottsDanielTascheManfredParameter estimation for exponential sums by approximate prony methodSignal Processing20109051631164210.1016/j.sigpro.2009.11.012 – reference: George A. Baker and Peter Graves-Morris. Padé Approximants. Cambridge University Press, Cambridge England, New York, 2nd edition, 1996. – reference: BatenkovDmitryYomdinYosefOn the accuracy of solving confluent Prony systemsSIAM Journal on Applied Mathematics2013731134154303314310.1137/110836584 – reference: LaurentMoniqueMourrainBernardA generalized flat extension theorem for moment matricesArchiv der Mathematik20099318798252064710.1007/s00013-009-0007-6 – reference: MourrainBernardIsolated points, duality and residuesJ. of Pure and Applied Algebra1996117&11846949314578510896.13020 – reference: PellerVladimir VAn excursion into the theory of Hankel operatorsHolomorphic spaces (Berkeley, CA, 1995), Math. Sci. Res. Inst. Publ1998336512016306460998.47016 – reference: KunisStefanPeterThomasRömerTimvon der OheUlrichA multivariate generalization of Prony’s methodLinear Algebra and its Applications20164903147342903110.1016/j.laa.2015.10.023 – reference: PlonkaGerlindTascheManfredProny methods for recovery of structured functionsGAMM-Mitteilungen2014372239258328502210.1002/gamm.201410011 – reference: SauerTomasProny’s method in several variablesNumerische Mathematik20171362411438364809310.1007/s00211-016-0844-8 – reference: PottsDanielTascheManfredParameter estimation for multivariate exponential sumsElectronic Transactions on Numerical Analysis20134020422430815701305.65093 – reference: CuytAnnieHow well can the concept of Padé approximant be generalized to the multivariate case?Journal of Computational and Applied Mathematics19991051–22550169057710.1016/S0377-0427(99)00028-X – reference: LasserreJean-BernardLaurentMoniqueMourrainBernardRostalskiPhilippTrébuchetPhilippeMoment matrices, border bases and real radical computationJournal of Symbolic Computation2013516385300578210.1016/j.jsc.2012.03.007 – reference: PowerStephen CFinite rank multivariable Hankel formsLinear Algebra and its Applications19824823724468322110.1016/0024-3795(82)90110-0 – reference: GolubGenePereyraVictorSeparable nonlinear least squares: The variable projection method and its applicationsInverse Problems2003192R1R26199178610.1088/0266-5611/19/2/201 – reference: RiquierCharlesLes systèmes d’équations aux dérivées partielles1910ParisGauthier-Villars40.0411.01 – reference: YangZaiXieLihuaStoicaPetreVandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-ResolutionIEEE Transactions on Information Theory201662636853701350675710.1109/TIT.2016.2553041 – reference: MasseyJamesShift-register synthesis and BCH decodingIEEE transactions on Information Theory196915112212724255610.1109/TIT.1969.1054260 – reference: AnderssonFredrikCarlssonMarcusde HoopMaarten VNonlinear approximation of functions in two dimensions by sums of wave packetsAppl. Comput. Harmon. Anal.2010292198213265245810.1016/j.acha.2009.09.001 – reference: Victor Pereyra and Godela Scherer, editors. Exponential Data Fitting and Its Applications. Bentham Science Publisher, 2012. – reference: Bernard Mourrain. A new criterion for normal form algorithms. In M. Fossorier, H. Imai, Shu Lin, and A. Poli, editors, Proc. AAECC, volume 1719 of LNCS, pages 430–443. Springer, Berlin, 1999. – reference: BeckermannBernhardGolubGene HLabahnGeorgeOn the numerical condition of a generalized Hankel eigenvalue problemNumerische Mathematik200710614168228600610.1007/s00211-006-0054-x – reference: SakataShojiroFinding a minimal set of linear recurring relations capable of generating a given finite two-dimensional arrayJournal of Symbolic Computation19885332133794658710.1016/S0747-7171(88)80033-6 – reference: Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and Its Applications, volume 149, pages 157–270. Springer, New York, 2009. – reference: BeckermannBernhardLabahnGeorgeA uniform approach for the fast computation of matrix-type Padé approximantsSIAM Journal on Matrix Analysis and Applications1994153804823128269610.1137/S0895479892230031 – reference: GiesbrechtMarkLabahnGeorgeLeeWen-shinSymbolic-numeric sparse interpolation of multivariate polynomialsJ. Symb. Comput.2009448943959252376110.1016/j.jsc.2008.11.003 – reference: PeterThomasPlonkaGerlindA generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operatorsInverse Problems2013292025001301196410.1088/0266-5611/29/2/025001 – reference: IarrobinoAnthonyKanevVassilPower Sums, Gorenstein Algebras, and Determinantal Loci1999BerlinSpringer10.1007/BFb0093426 – reference: MourrainBernardPanVictor YMultivariate Polynomials, Duality, and Structured MatricesJournal of Complexity2000161110180176240110.1006/jcom.1999.0530 – reference: Wolfgang Gröbner. Über das Macaulaysche inverse System und dessen Bedeutung für die Theorie der linearen Differentialgleichungen mit konstanten Koeffizienten. In Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg, volume 12, Issue 1, pages 127–132. Springer, 1937. – reference: MacWilliamsJessie FSloaneNeil J AThe Theory of Error-Correcting Codes, Volume 161977AmsterdamNorth Holland Publishing Co.0369.94008 – reference: MalgrangeBernardExistence et approximation des solutions des équations aux dérivées partielles et des équations de convolutionAnnales de l’institut Fourier195662713558699010.5802/aif.65 – reference: RochbergRichardToeplitz and Hankel operators on the Paley–Wiener spaceIntegral Equations and Operator Theory198710218723587824610.1007/BF01199078 – reference: CoxDavid ALittleJohnO’SheaDonalIdeals, Varieties, and Algorithms1992BerlinSpringer10.1007/978-1-4757-2181-2 – reference: James Joseph Sylvester. Essay on Canonical Form. The collected mathematical papers of J. J. Sylvester, Vol. I, Paper 34, Cambridge University Press. 1909 (XV und 688). G. Bell, London, 1851. – reference: CandèsEmmanuel JRombergJustinTaoTerenceStable signal recovery from incomplete and inaccurate measurementsCommunications on Pure and Applied Mathematics200659812071223223084610.1002/cpa.20124 – reference: Riche de PronyBaron GaspardEssai expérimental et analytique: sur les lois de la dilatabilité de fluides élastiques et sur celles de la force expansive de la vapeur de l’alcool, à différentes températuresJ. École Polytechnique179512476 – reference: The MUSIC AlgorithmA. Lee Swindlehurst and Thomas Kailath. A Performance Analysis of Subspace-Based Methods in the Presence of Model Errors - Part IIEEE Trans. on Signal Processing1992401758177410.1109/78.143447 – reference: HakopianHakop ATonoyanMariam GPartial differential analogs of ordinary differential equations and systemsNew York J. Math2004108911620523671073.35182 – reference: SchwartzLaurentThéorie des distributions1966ParisEditions Hermann0149.09501 – reference: BeylkinGregoryMonzónLucasOn approximation of functions by exponential sumsAppl. Comput. Harmon. Anal.20051911748214706010.1016/j.acha.2005.01.003 – reference: N. E. Golyandina and K. D. Usevich. 2D-Extension of Singular Spectrum Analysis: Algorithm and Elements of Theory. In Matrix Methods: Theory, Algorithms and Applications, pages 449–473. World Scientific Publishing, 2010. – reference: Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM ’79, pages 216–226, London, UK, 1979. Springer-Verlag. – reference: von zur GathenJoachimGerhardJürgenModern Computer Algebra20133CambridgeCambridge University Press10.1017/CBO9781139856065 – reference: Michael Ben-Or and Prasson Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In Proceedings of the twentieth annual ACM symposium on Theory of computing, STOC ’88, pages 301–309, New York, NY, USA, 1988. ACM. – reference: BerlekampElwyn RNonbinary BCH decodingIEEE Transactions on Information Theory196814224224210.1109/TIT.1968.1054109 – reference: Naum I. Ahiezer and Mark G. Kreĭn. Some Questions in the Theory of Moments. American Mathematical Society, Providence, 1962. – reference: ElkadiMohamedMourrainBernardIntroduction à la résolution des systèmes polynomiaux, Mathématiques & Applications2007BerlinSpringer10.1007/978-3-540-71647-1 – reference: Stef Graillat and Philippe Trébuchet. A new algorithm for computing certified numerical approximations of the roots of a zero-dimensional system. In Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pages 167–174. ACM, 2009. – reference: EisunbudDavidCommutative Algebra: With a View toward Algebraic Geometry, Graduate Texts in Mathematics1994BerlinSpringer – reference: CaixingGuFinite rank Hankel operators on the polydiskLinear Algebra and its Applications1999288269281167057910.1016/S0024-3795(98)10223-9 – reference: BernardiAlessandraBrachatJérômeComonPierreMourrainBernardGeneral tensor decomposition, moment matrices and applicationsJournal of Symbolic Computation2013525171301812810.1016/j.jsc.2012.05.012 – reference: RoyR.KailathT.ESPRIT-estimation of signal parameters via rotational invariance techniquesIEEE Transactions on Acoustics, Speech, and Signal Processing198937798499510.1109/29.32276 – reference: PedersenPaul SBasis for Power Series Solutions to Systems of Linear, Constant Coefficient Partial Differential EquationsAdvances in Mathematics19991411155166166714910.1006/aima.1998.1782 – reference: Jérémy Berthomieu, Brice Boyer, and Jean-Charles Faugère. Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences. In International Symposium on Symolic and Algebraic Compution, pages 61–68. ACM Press, 2015. – reference: Francis S. Macaulay. The Algebraic Theory of Modular Systems. Cambridge University Press, 1916. – reference: EmsalemJacquesGéométrie des points épaisBulletin de la S.M.F.19781063994160396.13017 – reference: Leopold Kronecker. Zur Theorie der Elimination Einer Variabeln aus Zwei Algebraischen Gleichungen. pages 535–600, 1880. – reference: BrachatJéromeComonPierreMourrainBernardTsigaridasEliasSymmetric tensor decompositionLinear Algebra and Applications201043318511872273610310.1016/j.laa.2010.06.046 – reference: BarachartLaurentSur la réalisation de Nerode des systèmes multi-indicielsC. R. Acad. Sc. Paris1984301715718 – reference: FitzpatrickPatrickNortonGraham HFinding a basis for the characteristic ideal of an n-dimensional linear recurring sequenceIEEE Transactions on Information Theory199036614801487108083610.1109/18.59953 – reference: HormanderLarsAn Introduction to Complex Analysis in Several Variables19903AmsterdamNorth Holland0685.32001 – reference: Fredrik Andersson and Marcus Carlsson. On the structure of positive semi-definite finite rank general domain Hankel and Toeplitz operators in several variables. Complex Analysis and Operator Theory, 11(4):755–784, 2017. – reference: Encyclopedia of Mathematics, April 2016. – reference: Constantin Carathéodory and Lipòt Fejér. Über den Zusammenhang der Extremen von Harmonischen Funktionen mit Ihren Koeffizienten und Über den Picard-Landauschen Satz. In Rendiconti del Circolo Matematico di Palermo (1884-1940), volume 32, pages 218–239. 1911. – reference: CurtoRaul EFialkowLawrence ASolution of the Truncated Complex Moment Problem for Flat Data1996Providence, R.IAmer Mathematical Society0876.30033 – reference: LimLek-HengComonPierreBlind multilinear identificationIEEE Transactions on Information Theory201460212601280316497410.1109/TIT.2013.2291876 – reference: AnderssonFredrikCarlssonMarcusOn General Domain Truncated Correlation and Convolution Operators with Finite RankIntegral Equations and Operator Theory2015823339370335578410.1007/s00020-015-2217-6 – reference: OberstUlrichPauerFranzThe Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant CoefficientsMultidimensional Systems and Signal Processing2001123–4253308186881110.1023/A:1011901522520 – reference: Bernard Mourrain and Philippe Trébuchet. Generalized normal forms and polynomials system solving. In M. Kauers, editor, Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’05), pages 253–260, 2005. – reference: FischerErnstÜber das Carathéodory’sche Problem, Potenzreihen mit positivem reellen Teil betreffendRendiconti del Circolo Matematico di Palermo (1884–1940)191132124025610.1007/BF03014797 – volume: 19 start-page: R1 issue: 2 year: 2003 ident: 9372_CR30 publication-title: Inverse Problems doi: 10.1088/0266-5611/19/2/201 – ident: 9372_CR33 doi: 10.1007/BF02948939 – ident: 9372_CR68 – volume: 15 start-page: 804 issue: 3 year: 1994 ident: 9372_CR10 publication-title: SIAM Journal on Matrix Analysis and Applications doi: 10.1137/S0895479892230031 – volume: 288 start-page: 269 year: 1999 ident: 9372_CR34 publication-title: Linear Algebra and its Applications doi: 10.1016/S0024-3795(98)10223-9 – volume: 433 start-page: 1851 year: 2010 ident: 9372_CR16 publication-title: Linear Algebra and Applications doi: 10.1016/j.laa.2010.06.046 – ident: 9372_CR71 doi: 10.1007/3-540-09519-5_73 – ident: 9372_CR63 doi: 10.1109/29.32276 – volume: 73 start-page: 134 issue: 1 year: 2013 ident: 9372_CR8 publication-title: SIAM Journal on Applied Mathematics doi: 10.1137/110836584 – volume: 33 start-page: 65 year: 1998 ident: 9372_CR54 publication-title: Holomorphic spaces (Berkeley, CA, 1995), Math. Sci. Res. Inst. Publ – volume: 12 start-page: 253 issue: 3–4 year: 2001 ident: 9372_CR52 publication-title: Multidimensional Systems and Signal Processing doi: 10.1023/A:1011901522520 – volume: 90 start-page: 1631 issue: 5 year: 2010 ident: 9372_CR58 publication-title: Signal Processing doi: 10.1016/j.sigpro.2009.11.012 – volume: 1 start-page: 24 year: 1795 ident: 9372_CR22 publication-title: J. École Polytechnique – ident: 9372_CR44 doi: 10.3792/chmm/1263317740 – ident: 9372_CR31 – volume: 117&118 start-page: 469 year: 1996 ident: 9372_CR48 publication-title: J. of Pure and Applied Algebra – ident: 9372_CR14 doi: 10.1145/2755996.2756673 – volume-title: Ideals, Varieties, and Algorithms year: 1992 ident: 9372_CR19 doi: 10.1007/978-1-4757-2181-2 – volume: 29 start-page: 025001 issue: 2 year: 2013 ident: 9372_CR56 publication-title: Inverse Problems doi: 10.1088/0266-5611/29/2/025001 – ident: 9372_CR41 doi: 10.1007/978-0-387-09686-5_7 – volume: 490 start-page: 31 year: 2016 ident: 9372_CR39 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2015.10.023 – ident: 9372_CR1 – ident: 9372_CR55 doi: 10.2174/97816080504821100101 – volume: 14 start-page: 242 issue: 2 year: 1968 ident: 9372_CR12 publication-title: IEEE Transactions on Information Theory doi: 10.1109/TIT.1968.1054109 – volume-title: Modern Computer Algebra year: 2013 ident: 9372_CR69 doi: 10.1017/CBO9781139856065 – volume: 29 start-page: 198 issue: 2 year: 2010 ident: 9372_CR5 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2009.09.001 – volume: 93 start-page: 87 issue: 1 year: 2009 ident: 9372_CR42 publication-title: Archiv der Mathematik doi: 10.1007/s00013-009-0007-6 – ident: 9372_CR51 doi: 10.1145/1073884.1073920 – volume: 5 start-page: 321 issue: 3 year: 1988 ident: 9372_CR64 publication-title: Journal of Symbolic Computation doi: 10.1016/S0747-7171(88)80033-6 – ident: 9372_CR11 doi: 10.1145/62212.62241 – volume: 32 start-page: 240 issue: 1 year: 1911 ident: 9372_CR26 publication-title: Rendiconti del Circolo Matematico di Palermo (1884–1940) doi: 10.1007/BF03014797 – volume: 6 start-page: 271 year: 1956 ident: 9372_CR46 publication-title: Annales de l’institut Fourier doi: 10.5802/aif.65 – ident: 9372_CR2 doi: 10.1090/mmono/002 – volume: 40 start-page: 1758 year: 1992 ident: 9372_CR67 publication-title: IEEE Trans. on Signal Processing doi: 10.1109/78.143447 – ident: 9372_CR32 doi: 10.1145/1576702.1576727 – ident: 9372_CR49 doi: 10.1007/3-540-46796-3_41 – volume-title: An Introduction to Complex Analysis in Several Variables year: 1990 ident: 9372_CR36 – volume: 106 start-page: 399 year: 1978 ident: 9372_CR25 publication-title: Bulletin de la S.M.F. – volume-title: Introduction à la résolution des systèmes polynomiaux, Mathématiques & Applications year: 2007 ident: 9372_CR24 doi: 10.1007/978-3-540-71647-1 – volume-title: Les systèmes d’équations aux dérivées partielles year: 1910 ident: 9372_CR61 – volume: 301 start-page: 715 year: 1984 ident: 9372_CR7 publication-title: C. R. Acad. Sc. Paris – volume: 48 start-page: 237 year: 1982 ident: 9372_CR60 publication-title: Linear Algebra and its Applications doi: 10.1016/0024-3795(82)90110-0 – volume: 60 start-page: 1260 issue: 2 year: 2014 ident: 9372_CR43 publication-title: IEEE Transactions on Information Theory doi: 10.1109/TIT.2013.2291876 – volume: 136 start-page: 411 issue: 2 year: 2017 ident: 9372_CR65 publication-title: Numerische Mathematik doi: 10.1007/s00211-016-0844-8 – volume: 40 start-page: 204 year: 2013 ident: 9372_CR59 publication-title: Electronic Transactions on Numerical Analysis – volume: 16 start-page: 110 issue: 1 year: 2000 ident: 9372_CR50 publication-title: Journal of Complexity doi: 10.1006/jcom.1999.0530 – volume: 10 start-page: 187 issue: 2 year: 1987 ident: 9372_CR62 publication-title: Integral Equations and Operator Theory doi: 10.1007/BF01199078 – volume: 44 start-page: 943 issue: 8 year: 2009 ident: 9372_CR29 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2008.11.003 – volume: 10 start-page: 89 year: 2004 ident: 9372_CR35 publication-title: New York J. Math – volume: 106 start-page: 41 issue: 1 year: 2007 ident: 9372_CR9 publication-title: Numerische Mathematik doi: 10.1007/s00211-006-0054-x – volume-title: Solution of the Truncated Complex Moment Problem for Flat Data year: 1996 ident: 9372_CR20 – ident: 9372_CR18 doi: 10.1007/BF03014796 – volume-title: Commutative Algebra: With a View toward Algebraic Geometry, Graduate Texts in Mathematics year: 1994 ident: 9372_CR23 – volume: 94 start-page: 231 year: 1970 ident: 9372_CR28 publication-title: Bulletin des Sciences Mathématiques. Deuxième Série – volume: 52 start-page: 51 year: 2013 ident: 9372_CR13 publication-title: Journal of Symbolic Computation doi: 10.1016/j.jsc.2012.05.012 – volume-title: The Theory of Error-Correcting Codes, Volume 16 year: 1977 ident: 9372_CR45 – ident: 9372_CR4 doi: 10.1007/s11785-016-0596-6 – volume: 141 start-page: 155 issue: 1 year: 1999 ident: 9372_CR53 publication-title: Advances in Mathematics doi: 10.1006/aima.1998.1782 – volume: 37 start-page: 239 issue: 2 year: 2014 ident: 9372_CR57 publication-title: GAMM-Mitteilungen doi: 10.1002/gamm.201410011 – volume: 36 start-page: 1480 issue: 6 year: 1990 ident: 9372_CR27 publication-title: IEEE Transactions on Information Theory doi: 10.1109/18.59953 – volume: 51 start-page: 63 year: 2013 ident: 9372_CR40 publication-title: Journal of Symbolic Computation doi: 10.1016/j.jsc.2012.03.007 – volume-title: Théorie des distributions year: 1966 ident: 9372_CR66 – ident: 9372_CR6 doi: 10.1017/CBO9780511530074 – volume: 59 start-page: 1207 issue: 8 year: 2006 ident: 9372_CR17 publication-title: Communications on Pure and Applied Mathematics doi: 10.1002/cpa.20124 – volume-title: Power Sums, Gorenstein Algebras, and Determinantal Loci year: 1999 ident: 9372_CR37 doi: 10.1007/BFb0093426 – volume: 19 start-page: 17 issue: 1 year: 2005 ident: 9372_CR15 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2005.01.003 – volume: 82 start-page: 339 issue: 3 year: 2015 ident: 9372_CR3 publication-title: Integral Equations and Operator Theory doi: 10.1007/s00020-015-2217-6 – volume: 105 start-page: 25 issue: 1–2 year: 1999 ident: 9372_CR21 publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/S0377-0427(99)00028-X – ident: 9372_CR38 – volume: 15 start-page: 122 issue: 1 year: 1969 ident: 9372_CR47 publication-title: IEEE transactions on Information Theory doi: 10.1109/TIT.1969.1054260 – volume: 62 start-page: 3685 issue: 6 year: 2016 ident: 9372_CR70 publication-title: IEEE Transactions on Information Theory doi: 10.1109/TIT.2016.2553041
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Snippet	We analyze the decomposition problem of multivariate polynomial–exponential functions from their truncated series and present new algorithms to compute their... We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their...
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SubjectTerms	Algebra Algebraic Geometry Algorithms Applications of Mathematics Computer Science Convolution Decomposition Economics Eigenvectors Exponential functions Hankel matrices Interpolation Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Operators (mathematics) Partial differential equations Polynomials Power series Theorems
Title	Polynomial–Exponential Decomposition From Moments
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