Moment Problems And Subsequences Of Moment Sequences
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| Author | : Saroj Aryal |
| Publisher | : |
| Total Pages | : 35 |
| Release | : 2011 |
| Genre | : Moment problems (Mathematics) |
| ISBN | : 9781267173195 |
The well-known theorems of Stieltjes, Hamburger and Hausdorff establish conditions on infinite sequences of real numbers to be the moments corresponding to functions in various spaces. Further works by Carathéodory, Schur and Nevanlinna connect moment problems to problems in function theory. In many problems associated with realization of a signal or an image, data may be corrupted or missing. Reconstruction of a function from moment sequences with missing elements is an interesting problem leading to several advances in image and/or signal reconstruction. It is easy to show that a subsequence of a moment sequence may not be a moment sequence. Conditions are obtained to show how rigid the space of sub-moment sequences of a moment sequence is. Some characteristics of the spaces of functions reconstructed from sub-moment problems are explored.
| Author | : Saroj Aryal |
| Publisher | : |
| Total Pages | : 73 |
| Release | : 2013 |
| Genre | : Moment problems (Mathematics) |
| ISBN | : 9781303423789 |
The well-known theorems of Stieltjes, Hamburger and Hausdorff establish conditions on infinite sequences of real numbers to be moment sequences. Further works by Carathéodory, Schur and Nevanlinna connect moment problems to problems in function theory and functions belonging to various spaces. In many problems associated with realization of a signal or an image, data may be corrupted or missing. Reconstruction of a function from moment sequences with missing terms is an interesting problem leading to advances in image and/or signal reconstruction. It is easy to show that a subsequence of a moment sequence may not be a moment sequence. Conditions are obtained to show how rigid the space of sub-moment sequences is and necessary and sufficient conditions for a sequence to be a sub-moment sequence are established. A deep connection between the sub-moment measures and the moment measures is derived and the determinacy of the moment and sub-moment problems are related. This problem is further related to completion of positive Hankel matrices. Data obtained from physical experiments and natural processes are often moment multisequences. Conditions are established for a subsequence of a moment multisequence to be positive. Furthermore, subsequences of a moment multisequence that are also moment multisequences are fully characterized.
| Author | : Konrad Schmüdgen |
| Publisher | : Springer |
| Total Pages | : 530 |
| Release | : 2017-11-09 |
| Genre | : Mathematics |
| ISBN | : 3319645463 |
This advanced textbook provides a comprehensive and unified account of the moment problem. It covers the classical one-dimensional theory and its multidimensional generalization, including modern methods and recent developments. In both the one-dimensional and multidimensional cases, the full and truncated moment problems are carefully treated separately. Fundamental concepts, results and methods are developed in detail and accompanied by numerous examples and exercises. Particular attention is given to powerful modern techniques such as real algebraic geometry and Hilbert space operators. A wide range of important aspects are covered, including the Nevanlinna parametrization for indeterminate moment problems, canonical and principal measures for truncated moment problems, the interplay between Positivstellensätze and moment problems on semi-algebraic sets, the fibre theorem, multidimensional determinacy theory, operator-theoretic approaches, and the existence theory and important special topics of multidimensional truncated moment problems. The Moment Problem will be particularly useful to graduate students and researchers working on moment problems, functional analysis, complex analysis, harmonic analysis, real algebraic geometry, polynomial optimization, or systems theory. With notes providing useful background information and exercises of varying difficulty illustrating the theory, this book will also serve as a reference on the subject and can be used for self-study.
| Author | : Fernanda Botelho |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 250 |
| Release | : 2017-04-18 |
| Genre | : Mathematics |
| ISBN | : 1470427729 |
This volume contains the proceedings of the Workshop on Problems and Recent Methods in Operator Theory, held at the University of Memphis, Memphis, TN, from October 15–16, 2015 and the AMS Special Session on Advances in Operator Theory and Applications, in Memory of James Jamison, held at the University of Memphis, Memphis, TN, from October 17–18, 2015. Operator theory is at the root of several branches of mathematics and offers a broad range of challenging and interesting research problems. It also provides powerful tools for the development of other areas of science including quantum theory, physics and mechanics. Isometries have applications in solid-state physics. Hermitian operators play an integral role in quantum mechanics very much due to their “nice” spectral properties. These powerful connections demonstrate the impact of operator theory in various branches of science. The articles in this volume address recent problems and research advances in operator theory. Highlighted topics include spectral, structural and geometric properties of special types of operators on Banach spaces, with emphasis on isometries, weighted composition operators, multi-circular projections on function spaces, as well as vector valued function spaces and spaces of analytic functions. This volume gives a succinct overview of state-of-the-art techniques from operator theory as well as applications to classical problems and long-standing open questions.
| Author | : Hayoung Choi |
| Publisher | : |
| Total Pages | : 56 |
| Release | : 2015 |
| Genre | : Moment problems (Mathematics) |
| ISBN | : 9781339054636 |
Moment problems arise naturally in many areas of Mathematics, Economics and Operations research. While moment problems have numerous applications to extremal problems, optimization and limit theorems in probability theory, they rely on a complete set of moments or truncated moment sequences. Due to missing moment entries or availability of truncated moment sequences, sometimes we need to work in the space of incomplete moment sequences. Moment problems with missing entries are closely related to Hankel matrix completion problems. In this dissertation we give solutions to Hamburger moment problems with missing entries. The problem of completing partial positive sequences is considered. The main result is a characterization of positive (semi)definite completable patterns, namely patterns that guarantee the existence of a Hamburger moment completion of a partial positive (semi)definite sequence. Moreover, several patterns which are not positive definite completable are given. Furthermore, we characterize the determinate case by certain subsequence of given moment sequence. For a positive sequence if its subsequence with the pattern of arithmetic progression is determinate, then the sequence is determinate. Also, if a sequence is indeterminate then its subsequences with pattern of arithmetic progression are indeterminate. In the last part of this thesis, we apply moment completion problems to reconstruct Radon transform with missing data. Radon transform is a well known tool for reconstructing data from its projections. Reconstruction of Radon transform with missing data is closely related to reconstruction of a function from moment sequences with missing terms. A new range theorem is established for the Radon transform based on the Hamburger moment problem in two variables, and the sparse moment problem is converted into the Radon transform with missing data and vice versa. A modified Radon transform is introduced and its inversion formula is established.
| Author | : James Alexander Shohat |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 160 |
| Release | : 1943-12-31 |
| Genre | : Mathematics |
| ISBN | : 0821815016 |
The book was first published in 1943 and then was reprinted several times with corrections. It presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. In addition to initial developments by Stieltjes, Markov, and Chebyshev, later contributions by Hamburger, Nevanlinna, Hausdorff, Stone, and others are discussed. The book also contains some results on the trigonometric moment problem and a chapter devoted to approximate quadrature formulas.
| Author | : Mark Grigorʹevich Kreĭn |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 430 |
| Release | : |
| Genre | : Mathematics |
| ISBN | : 9780821886717 |
In this book, an extensive circle of questions originating in the classical work of P. L. Chebyshev and A. A. Markov is considered from the more modern point of view. It is shown how results and methods of the generalized moment problem are interlaced with various questions of the geometry of convex bodies, algebra, and function theory. From this standpoint, the structure of convex and conical hulls of curves is studied in detail and isoperimetric inequalities for convex hulls are established; a theory of orthogonal and quasiorthogonal polynomials is constructed; problems on limiting values of integrals and on least deviating functions (in various metrics) are generalized and solved; problems in approximation theory and interpolation and extrapolation in various function classes (analytic, absolutely monotone, almost periodic, etc.) are solved, as well as certain problems in optimal control of linear objects.
| Author | : James Alexander Shohat |
| Publisher | : American Mathematical Society(RI) |
| Total Pages | : 168 |
| Release | : 1950 |
| Genre | : Mathematics |
| ISBN | : |
Presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. This book discusses the initial developments by Stieltjes, Markov, and Chebyshev, and later contributions by Hamburger, Nevanlinna, Hausdorff, and Stone.
| Author | : Raúl E. Curto |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 69 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 0821804855 |
We introduce a matricial approach to the truncated complex moment problem, and apply it to the case of moment matrices of flat data type, for which the columns corresponding to the homogeneous monomials in [italic]z and [italic]z̄ of highest degree can be written in terms of monomials of lower degree. We discuss the connection between complex moment problems and the subnormal completion problem for 2-variable weighted shifts, and present in detail the construction of solutions for truncated complex moment problems associated with monomials of degrees one and two.
| Author | : Viktor Benes |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 311 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 9401155321 |
The last decade has seen a remarkable development of the "Marginal and Moment Problems" as a research area in Probability and Statistics. Its attractiveness stemmed from its lasting ability to provide a researcher with difficult theoretical problems that have direct consequences for appli cations outside of mathematics. The relevant research aims centered mainly along the following lines that very frequently met each other to provide sur prizing and useful results : -To construct a probability distribution (to prove its existence, at least) with a given support and with some additional inner stochastic property defined typically either by moments or by marginal distributions. -To study the geometrical and topological structure of the set of prob ability distributions generated by such a property mostly with the aim to propose a procedure that would result in a stochastic model with some optimal properties within the set of probability distributions. These research aims characterize also, though only very generally, the scientific program of the 1996 conference "Distributions with given marginals and moment problems" held at the beginning of September in Prague, Czech Republic, to perpetuate the tradition and achievements of the closely related 1990 Roma symposium "On Frechet Classes" 1 and 1993 Seattle" AMS Summer Conference on Marginal Problem".
