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Search Results: 1 - 10 of 1319 matches for " Mihai Putinar "
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A Striktpositivstellensatz for measurable functions (corrected version)
Mihai Putinar
Mathematics , 2009,
Abstract: A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous result for polynomials or certain rational functions was amply exploited during the last decade in a variety of applications.
Extremal solutions of the two-dimensional $L$-problem of moments, II
Mihai Putinar
Mathematics , 1995,
Abstract: All extremal solutions of the truncated $L$-problem of moments in two real variables , with support contained in a given compact set, are described as characteristic functions of semi-algebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal function of a naturally associated hyponormal operator with rank-one self-commutator, provides a natural defining function for these semi-algebraic sets. We find an intrinsic characterization of this kernel and we describe a series of analytic continuation properties of it which are closely related to the behaviour of the Schwarz reflection function in portions of the boundary of the extremal supporting set.
Quillen property of real algebraic varieties
Mihai Putinar,Claus Scheiderer
Mathematics , 2013,
Abstract: Let I be a conjugation-invariant ideal in the complex polynomial ring with variables z_1,...,z_n and their conjugates. The ideal I has the Quillen property if every real valued, strictly positive polynomial on the real zero set of I in C^n is a sum of hermitian squares modulo I. We first relate the Quillen property to the archimedean property from real algebra. Using hereditary calculus, we then quantize and show that the Quillen property implies the subnormality of commuting tuples of Hilbert space operators satisfying the identities in I. In the finite rank case we give a complete geometric characterization of when the identities in $I$ imply normality for a commuting tuple of matrices. This geometric interpretation provides simple means to refute Quillen's property of an ideal. We also generalize these notions and results from real algebraic sets to semi-algebraic sets in C^n.
A relative Grace Theorem for complex polynomials
Daniel Plaumann,Mihai Putinar
Mathematics , 2014,
Abstract: We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.
On the classification of complex vector bundles of stable rank
Constantin Bǎnicǎ,Mihai Putinar
Mathematics , 2006,
Abstract: One describes, using a detailed analysis of Atiyah--Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle.
Multivariate Determinateness
Mihai Putinar,Konrad Schmüdgen
Mathematics , 2008,
Abstract: The uniqueness question of the multivariate moment problem is studied by different methods: Hilbert space operators, complex function theory, polynomial approximation, disintegration, integral geometry. Most of the known results in the multi-dimensional case are reviewed and reproved, and a number of new determinacy criteria are developed.
Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
John William Helton,Mihai Putinar
Mathematics , 2006,
Abstract: 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.
Conditioning moments of singular measures for entropy optimization. I
Marko Budi?i?,Mihai Putinar
Mathematics , 2012, DOI: 10.1016/j.indag.2012.05.008
Abstract: In order to process a potential moment sequence by the entropy optimization method one has to be assured that the original measure is absolutely continuous with respect to Lebesgue measure. We propose a non-linear exponential transform of the moment sequence of any measure, including singular ones, so that the entropy optimization method can still be used in the reconstruction or approximation of the original. The Cauchy transform in one variable, used for this very purpose in a classical context by A.\ A.\ Markov and followers, is replaced in higher dimensions by the Fantappi\`{e} transform. Several algorithms for reconstruction from moments are sketched, while we intend to provide the numerical experiments and computational aspects in a subsequent article. The essentials of complex analysis, harmonic analysis, and entropy optimization are recalled in some detail, with the goal of making the main results more accessible to non-expert readers. Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube domain; exponential transform
Conditioning moments of singular measures for entropy maximization II: Numerical examples
Marko Budi?i?,Mihai Putinar
Mathematics , 2014,
Abstract: If moments of singular measures are passed as inputs to the entropy maximization procedure, the optimization algorithm might not terminate. The framework developed in our previous paper demonstrated how input moments of measures, on a broad range of domains, can be conditioned to ensure convergence of the entropy maximization. Here we numerically illustrate the developed framework on simplest possible examples: measures with one-dimensional, bounded supports. Three examples of measures are used to numerically compare approximations obtained through entropy maximization with and without the conditioning step.
Positivity aspects of the Fantappie transform
J. E. McCarthy,Mihai Putinar
Mathematics , 2004,
Abstract: A Hilbert space approach to the classical Fantappie transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappie transforms of positive measures on the unit ball in $\C^n$ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators.
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