Dual finite frames for vector spaces over an arbitrary field with applications

Patricia Mariela Morillas

doi:10.52737/18291163-2021.13.2-1

Authors

Patricia Mariela Morillas Instituto de Matemática Aplicada San Luis (UNSL-CONICET)

DOI:

https://doi.org/10.52737/18291163-2021.13.2-1

Keywords:

Vector Spaces, Fields, Dual Frames, Hilbert Spaces, Metric Vector Spaces, Ultrametric Normed Vector Spaces

Abstract

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.

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