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Yuhjia Lee - One of the best experts on this subject based on the ideXlab platform.
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the Riesz Representation Theorem on infinite dimensional spaces and its applications
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2002Co-Authors: Yuhjia Lee, Chenyuh ShihAbstract:Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let be the collection of complex-valued continuous functions f defined on E* such that is finite for every m. Then is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz Representation Theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation Theorem on E* for γ≥1 and show that the space spanned by the class {exp[i(x,ξ)] : ξ ∈ E} is dense in for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.
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a generalization of the Riesz Representation Theorem to infinite dimensions
Journal of Functional Analysis, 1997Co-Authors: Yuhjia LeeAbstract:LetHbe a real separable Hilbert space and letE⊂Hbe a nuclear space with the chain of Hilbert spaces {Ep: p=1, 2, 3, …} such thatE=∩∞p=1 Ep. LetE* andE−pdenote the dual spaces ofEandEp, respectively. Let Cpbe the collection of real-valued functionsfdefined onE−psuch thatfis uniformly continuous on bounded subsets ofE−pand such that ‖f‖∞, p≔supx∈E−p{|f(x)| exp(−12 |x|2−p)} is finite. Set C∞=∩∞p=1 Cp. ThenC∞is a complete countably normed space equipped with the family {‖·‖∞, p: p=1, 2, 3, …} of norms. In this paper it is shown that to every bounded linear functionalFin C*∞, there corresponds a signed measureνFsuch thatF(ϕ)=∫E* ϕ(x) νF(dx) forϕ∈C∞. It is also shown that there exists somepsuch that the measurable support ofνis contained inE−pand ∫E−p exp(12 |x|2−p) |νF| (dx)<∞. This result extends the Riesz Representation Theorem to infinite dimensions. In the course of the proof, an infinite dimensional analogue of the Weierstrass approximation Theorem is also established onE*.
Heinz Konig - One of the best experts on this subject based on the ideXlab platform.
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new version of the daniell stone Riesz Representation Theorem
Positivity, 2008Co-Authors: Heinz KonigAbstract:The traditional Representation Theorems after Daniell-Stone and Riesz were in a kind of separate existence until Pollard-Topsoe 1975 and Topsoe 1976 were the first to put them under common roofs. In the same spirit the present article wants to obtain a unified Representation Theorem in the context of the author’s work in measure and integration. It is an inner Theorem like the previous ones. The basis is the recent comprehensive inner Daniell-Stone Theorem, so that in particular there are no a priori assumptions on the additive behaviour of the data.
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the daniell stone Riesz Representation Theorem
Operator theory, 1995Co-Authors: Heinz KonigAbstract:We present a new version of the Daniell-Stone Representation Theorem for certain lattice cones of [0, ∞[-valued functions. It contains a new version of the Riesz Representation Theorem on Hausdorff topological spaces. The latter result characterizes those elementary integrals, defined on certain lattice cones of upper semicontinuous [0, ∞[-valued functions concentrated on compact subsets, which come from Radon measures.
Walter Roth - One of the best experts on this subject based on the ideXlab platform.
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Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies
Positivity, 2002Co-Authors: Walter RothAbstract:On a given cone (resp. vector space) $$\mathcal{Q}$$ we consider an initial topology and order induced by a family of linear operators into a second cone $$\mathcal{P}$$ which carries a locally convex topology. We prove that monotone linear functionals on $$\mathcal{Q}$$ which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on $$\mathcal{P}$$ . Based on the Riesz Representation Theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces.
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a Riesz Representation Theorem for cone valued functions
Abstract and Applied Analysis, 1999Co-Authors: Walter RothAbstract:We consider Borel measures on a locally compact Hausdorff space whose values are linear functionals on a locally convex cone. We define integrals for cone-valued functions and verify that continuous linear functionals on certain spaces of continuous cone-valued functions endowed with an inductive limit topology may be represented by such integrals.
Hagverdiyev Ali - One of the best experts on this subject based on the ideXlab platform.
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Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation
Scholarly Commons, 2019Co-Authors: Abdulla Ugur, Goldfarb Jonathan, Hagverdiyev AliAbstract:Optimal control of coefficients in the free boundary problem for the second order parabolic PDE modeling biomedical engineering problem on the laser ablation of biological tissues is analyzed. Optimal control in Hilbert-Besov spaces framework is employed where coefficient of the PDE and free boundary are components of the control vector and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proved. Gradient descent algorithm based on Frechet differentiability in Hilbert-Besov spaces complemented with preconditioning or increase of regularity of the Frechet gradient through implementation of the Riesz Representation Theorem is implemented. Numerical results are demonstrated for the optimal control of the two-phase Stefan problem based on the optimize-then-discretize approach through implementation of the gradient method in Hilbert-Besov spaces, preconditioning, simultaneous and individual identification of control parameters, as well as sensitivity analysis with respect to initial data
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Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems
'Elsevier BV', 2018Co-Authors: Abdulla, Ugur G., Bukshtynov Vladislav, Hagverdiyev AliAbstract:This paper presents computational analysis of the inverse Stefan type free boundary problem, where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary. We pursue optimal control framework introduced in {\it U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340;\ 10, 4(2016), 869--898}, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the quadratic declinations from the available measurements of the temperature distribution at the final moment, phase transition temperature on the free boundary, and the final position of the free boundary. We develop gradient descent algorithm based on Frechet differentiability in Hilbert-Besov spaces complemented with preconditioning or increase of regularity of the Frechet gradient through implementation of the Riesz Representation Theorem. Three model examples with various levels of complexity are considered. Extensive comparative analysis through implementation of preconditioning and Tikhonov regularization, calibration of preconditioning and regularization parameters, effect of noisy data, comparison of simultaneous identification of control parameters vs. nested optimization is pursued.Comment: 31 pages, 15 figure
Javier Zuniga - One of the best experts on this subject based on the ideXlab platform.
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equilibrium problems and Riesz Representation Theorem
Research Papers in Economics, 2016Co-Authors: John Cotrina, Javier ZunigaAbstract:The purpose of this paper is to give an alternative proof of the Riesz Representation Theorem using the well-known Theorem of Ky Fan minimax inequality applied to equilibrium problems.
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equilibrium problems and Riesz Representation Theorem
Repositorio de la Universidad del Pacífico - UP, 2016Co-Authors: John Cotrina, Javier ZunigaAbstract:El proposito de este trabajo es mostrar una prueba alternativa del teorema de representacion de Riesz, mediante el uso del conocido teorema de la desigualdad de minimax de Ky Fan aplicado a problemas de equilibrio.URI
