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29
Orthogonal Polynomials of Several Variables
 Encyclopedia of Mathematics and its Applications
, 2001
"... Abstract. We report on the recent development on the general theory of orthogonal polynomials in several variables, in which results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation. 1 ..."
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Cited by 133 (28 self)
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Abstract. We report on the recent development on the general theory of orthogonal polynomials in several variables, in which results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation. 1
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
"... ..."
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 36 (6 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Solving Moment Problems By Dimensional Extension
, 1999
"... this paper is devoted to an analysis of moment problems in R ..."
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Cited by 25 (3 self)
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this paper is devoted to an analysis of moment problems in R
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 18 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
The Complex Moment Problem and Subnormality: A Polar Decomposition Approach
, 1998
"... . It has been known that positive definiteness does not guarantee for a bisequence to be a complex moment one. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. S ..."
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Cited by 16 (4 self)
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. It has been known that positive definiteness does not guarantee for a bisequence to be a complex moment one. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. So as to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context (Theorems 9 and 29). All this enables us to prove: characterizations of subnormality of unbounded operators having invariant domain (Theorems 37 and 39) and their further applications (Theorems 41 and 43), and description of complex moment problem on real algebraic curves (Theorems 52 and 56). The latter question was completed in Appendix in which we relate the complex moment problem to the twodimensional real one, with emphasis on real algebraic sets. 1991 Mathematics Subject Classification. Primary 44A60, 47B20, 43A35; Secondary 60B99. Key words and phrase...
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
Solving the Truncated Moment Problem Solves the Full Moment Problem
, 1999
"... It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem. ..."
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Cited by 4 (0 self)
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It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem.
Determinate multidimensional measures, the extended Carleman theorem and quasianalytic weights. The Annals of Probability
"... Abstract. We prove in a direct fashion that a multidimensional probability measure µ is determinate if the higher dimensional analogue of Carleman’s condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated ..."
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Cited by 4 (1 self)
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Abstract. We prove in a direct fashion that a multidimensional probability measure µ is determinate if the higher dimensional analogue of Carleman’s condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated Lpspaces for 1 ≤ p < ∞. In particular these three statements hold if the reciprocal of a quasianalytic weight has finite integral under µ. This in turn leads to a sufficient integral condition for the determinacy of multidimensional probability measures and the above density results. 1. Introduction and