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.
Highlights
Frames were introduced by Duffin and Schaeffer in 1952 for some Hilbert function spaces [20]
In order to study the representations of the elements in V provided by a frame, we introduce a concept of dual frame and analyze its properties
The theory of dual frames developed here gives the possibility to apply the representations provided by frames even in those areas where there is no other structure defined on the vector space
Summary
Frames were introduced by Duffin and Schaeffer in 1952 for some Hilbert function spaces [20]. The theory of dual frames developed here gives the possibility to apply the representations provided by frames even in those areas where there is no other structure defined on the vector space. It allows considering dual frames for different vector spaces with additional structures We apply this theory to study dual frames for three particular vector spaces. We study frames and dual frames for ultrametric normed vector spaces focusing principally on perturbations results As another application of the developed theory, the frame representation of operators and the solution of operator equations in a Petrov-Galerkin scheme are considered. We consider the representation of operators using dual frames and their application to the solution of operator equations
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