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

MORILLAS, PATRICIA MARIELA

Armenian Journal of Mathematics

Republic of Armenia National Academy of Sciences

Año: 2021 vol. 13 p. 1 - 36

In the present paper, we study frames for finitedimensional 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 f ields. 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.