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Butler Svetlana - One of the best experts on this subject based on the ideXlab platform.
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Deficient topological measures on locally compact spaces
2019Co-Authors: Butler SvetlanaAbstract:Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on compact sets, inner regular on open sets, and outer regular on closed sets. Deficient topological measures generalize measures and topological measures. First we investigate positive, negative, and total variation of a signed set function that is only assumed to be finitely additive on compact sets. These positive, negative, and total variations turn out to be deficient topological measures. Then we examine finite additivity, superadditivity, smoothness, and other properties of deficient topological measures. We obtain methods for generating new deficient topological measures. We provide necessary and sufficient conditions for a deficient topological measure to be a topological measure and to be a measure. The results presented are necessary for further study of topological measures, deficient topological measures, and corresponding non-linear functionals on locally compact spaces.Comment: 27 page
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Solid-set functions and topological measures on locally compact spaces
2019Co-Authors: Butler SvetlanaAbstract:A topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. Almost all works devoted to topological measures, corresponding non-linear functionals, and their applications deal with compact spaces. The present paper is one in a series that investigates topological measures and corresponding non-linear functionals on locally compact spaces. Here we examine solid and semi-solid sets on a locally compact space. We then give a method of constructing topological measures from solid-set functions on a locally compact, connected, locally connected space. The paper gives examples of finite and infinite topological measures on locally compact, non-compact spaces and presents an easy way to generate topological measures on spaces whose one-point compactification has genus 0 from existing examples of topological measures on compact spaces.Comment: 42 page
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Integration with respect to deficient topological measures on locally compact spaces
2019Co-Authors: Butler SvetlanaAbstract:Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative vanishing at infinity function; and it produces a signed deficient topological measure if we use a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure and Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure $ \mu$ that assumes finitely many values, there is a function $ f $ such that $\int_X f \, d \mu = 0$, but $\int_X (-f )\, d \mu \neq 0$. We present different criteria for $\int_X f \, d \mu = 0$. We also prove some convergence results, including a Monotone convergence theorem.Comment: 30 page
Arash Ghaani Farashahi - One of the best experts on this subject based on the ideXlab platform.
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a class of abstract linear representations for convolution function algebras over homogeneous spaces of compact groups
2018Co-Authors: Arash Ghaani FarashahiAbstract:This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let G be a compact group and H a closed subgroup of G . Let μ be the normalized G -invariant measure over the compact homogeneous space G/H associated with Weil's formula and 1≤p<∞ . We then present a structured class of abstract linear representations of the Banach convolution function algebras Lp(G/H,μ).
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abstract plancherel trace formulas over homogeneous spaces of compact groups
2017Co-Authors: Arash Ghaani FarashahiAbstract:This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let be a compact group and let be a closed subgroup of . Let be the left coset space of in and let be the normalized -invariant measure on associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space .
Mauricio Velasco - One of the best experts on this subject based on the ideXlab platform.
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approximate super resolution of positive measures in all dimensions
2020Co-Authors: Hernan Garcia, Camilo Hernandez, Mauricio Junca, Mauricio VelascoAbstract:Abstract We study the problem of reconstructing a positive discrete measure on a compact set K ⊆ R n from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.
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approximate super resolution of positive measures in all dimensions
2018Co-Authors: Hernan Garcia, Camilo Hernandez, Mauricio Junca, Mauricio VelascoAbstract:We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.
Yoshikata Kida - One of the best experts on this subject based on the ideXlab platform.
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measure equivalence rigidity of the mapping class group
2010Co-Authors: Yoshikata KidaAbstract:We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.
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orbit equivalence rigidity for ergodic actions of the mapping class group
2008Co-Authors: Yoshikata KidaAbstract:We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.
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measure equivalence rigidity of the mapping class group
2006Co-Authors: Yoshikata KidaAbstract:We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice embeddings of the mapping class group into a locally compact second countable group. We also obtain similar results for finite direct products of mapping class groups.
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the mapping class group from the viewpoint of measure equivalence theory
2005Co-Authors: Yoshikata KidaAbstract:We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.
Hernan Garcia - One of the best experts on this subject based on the ideXlab platform.
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approximate super resolution of positive measures in all dimensions
2020Co-Authors: Hernan Garcia, Camilo Hernandez, Mauricio Junca, Mauricio VelascoAbstract:Abstract We study the problem of reconstructing a positive discrete measure on a compact set K ⊆ R n from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.
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approximate super resolution of positive measures in all dimensions
2018Co-Authors: Hernan Garcia, Camilo Hernandez, Mauricio Junca, Mauricio VelascoAbstract:We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.