Extension procedures for lattice Lipschitz operators on Euclidean spaces
| Autor: | Roger Arnau, J. M. Calabuig, Ezgi Erdoğan, Enrique A. Sánchez Pérez |
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| Přispěvatelé: | Arnau R., Calabuig J., ERDOĞAN E., Sánchez Pérez E. A. |
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Eigenvalue
Temel Bilimler (SCI) Geometri ve Topoloji Diagonal Analiz MATHEMATICS Cebir ve Sayı Teorisi Euclidean space Extension MATEMATİK Uygulamalı matematik Bilgisayar Bilimleri Lipschitz Matematik Algebra and Number Theory Temel Bilimler MATEMATİK DİSİPLİNLERARASI UYGULAMALAR Applied Mathematics MATHEMATICS INTERDISCIPLINARY APPLICATIONS Hesaplamalı Matematik MATEMATİK UYGULAMALI Banach lattice Computational Mathematics MATHEMATICS APPLIED Fizik Bilimleri Computer Science Natural Sciences (SCI) Physical Sciences Operator Geometry and Topology Natural Sciences Analysis |
| Popis: | We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions. |
| Databáze: | OpenAIRE |
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