Open AccessDual finite frames for vector spaces over an arbitrary field with applications
Author(s) -
Patricia Mariela Morillas
Publication year - 2021
Publication title -
armenian journal of mathematics
Language(s) - English
Resource type - Journals
ISSN - 1829-1163
DOI - 10.52737/18291163-2021.13.2-1
Subject(s) - dual space , mathematics , vector space , dual pair , vector field , hilbert space , locally convex topological vector space , normed vector space , dual representation , dual (grammatical number) , topological tensor product , pure mathematics , duality (order theory) , ultrametric space , field (mathematics) , algebra over a field , topological vector space , interpolation space , space (punctuation) , metric space , computer science , functional analysis , topological space , operating system , art , biochemistry , chemistry , geometry , literature , gene
In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.