40 research outputs found
Symmetry in Mathematical Analysis and Functional Analysis
No full textThe present reprint provides some theoretical results (and their applications) in the fields of mathematical analysis and functional analysis, in which the concept of symmetry plays an essential role. More specifically, various problems are investigated in areas, such as: optimization problems, polynomial approximation on unbounded subsets, moment problems, variational inequalities, evolutionary problems, dynamical systems, generalized convexity, partial differential equations, and special spaces of self-adjoint operators. With various examples and applications to complement and substantiate the mathematical developments, the present reprint is a valuable guide for researchers, engineers, and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science, and economics
Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications
No full textThis paper starts by recalling the author’s results on polynomial approximation over a Cartesian product A of closed unbounded intervals and its applications to solving Markov moment problems. Under natural assumptions, the existence and uniqueness of the solution are deduced. The characterization of the existence of the solution is formulated by two inequalities, one of which involves only quadratic forms. This is the first aim of this work. Characterizing the positivity of a bounded linear operator only by means of quadratic forms is the second aim. From the latter point of view, one solves completely the difficulty arising from the fact that there exist nonnegative polynomials on ℝn, n≥2, which are not sums of squares
On the Moment Problem and Related Problems
No full textFirstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work
On Special Properties for Continuous Convex Operators and Related Linear Operators
No full textThis paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above
Approximation and the Multidimensional Moment Problem
No full textThe aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire R2, or on R+2, are considered. Such results are discussed in Lμ1R2 and in CS1×S2-type spaces, for a large class of measures, μ, for compact subsets Si, i=1,2 of the interval [0,+∞). Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined
Extension of Linear Operators and Polynomial Approximation, with Applications to Markov Moment Problem and Mazur-Orlicz Theorem
No full textOne recalls the relationship between the Markov moment problem and extension of linear functionals (or operators), with two constraints. One states necessary and sufficient conditions for the existence of solutions of some abstract vector-valued Markov moment problems, by means of a general Hahn-Banach principle. The classical moment problem is discussed as a particular important case. A short section is devoted to applications of polynomial approximation in studying the existence and uniqueness of the solutions for two types of Markov moment problems. Mazur-Orlicz theorem is also recalled and applied. We use general type results in studying related problems which involve concrete spaces of functions and self-adjoint operators. Sometimes, the uniqueness of the solution follows too. Most of our solutions are operator-valued or function-valued
Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems
No full textThe purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above
Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation
No full textWe start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval [0,+∞). Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved
Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms
No full textAs is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T1≤T≤T2 for the linear operator T, only in terms of quadratic forms, is deduced. Here, T1, T,and T2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples
Newton’s Method for Convex Operators and Applications
No full textThis review work presents the general statement of a variant of Newton’s method for convex monotone operators and its applications. We consider the estimation of the absolute error too. One makes the connection to the contraction principle. One of the applications is approximating where a positive selfadjoint operator is acting on a Hilbert space. One works with “global” convex monotone operators. For the local approach, we mention appropriate references.<br /
