Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems
The 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-p...
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| Veröffentlicht in: | Symmetry (Basel) Jg. 15; H. 7; S. 1471 |
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| Abstract | The 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. |
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| AbstractList | The 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. The 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[sub.+] 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. |
| Audience | Academic |
| Author | Olteanu, Octav |
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| Cites_doi | 10.1007/s00009-017-0870-5 10.1016/S0893-9659(03)00030-2 10.2307/2371063 10.1016/j.crma.2005.10.009 10.1016/j.jfa.2022.109674 10.1017/S0017089501030130 10.3390/sym13060986 10.1007/978-3-031-21502-5 10.1007/BF01446568 10.5802/afst.108 10.1007/s11785-023-01339-7 10.3390/sym14020398 10.1007/978-0-387-70914-7 10.1007/BF02392450 10.3390/math9040309 10.1090/S0002-9939-09-10016-3 10.3390/sym15010076 10.1090/S0002-9939-1991-1059628-5 10.1007/978-1-4612-1128-0 10.1007/978-81-322-2148-7 10.1007/BF01420423 10.1016/0022-1236(84)90042-9 10.1137/S0040585X97T990083 |
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| SubjectTerms | Algebra Analytic functions Approximation Asymmetry Complex variables Functional analysis Functional equations Half planes Interpolation Mathematical analysis Mathematical functions Matrix theory Optimization Polynomials Symmetry Theorems |
| Title | Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems |
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