Arnau Notari, Andrés Roger

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0000-0003-2544-8875
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https://www.upv.es/ficha-personal/ararnnot
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https://panorama.upv.es/es/ipublic/researcher/340763
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2022 - 20232
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Subject: Metric space ×

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Now showing 1 - 2 of 2
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    Publication
    Representation of Lipschitz Maps and Metric Coordinate Systems
    (MDPI AG, 2022-10) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; AGENCIA ESTATAL DE INVESTIGACION; UNIVERSIDAD POLITECNICA DE VALENCIA
    [EN] Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze ¿Pietsch Theorem inspired factorizations" through subspaces of `¿ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane¿Whitney formulas, are also given.
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    Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression
    (MDPI AG, 2023-04-07) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; AGENCIA ESTATAL DE INVESTIGACION; UNIVERSIDAD POLITECNICA DE VALENCIA
    [EN] This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p-average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p = 2, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given.