Frames and representing systems in Fr\'echet spaces and their duals

TL;DR

This paper explores the theory of frames and Bessel sequences in Fréchet spaces and their duals, analyzing their relation to Schauder frames and representing systems, with applications to analytic function spaces, sampling, and Dirichlet series.

Contribution

It introduces and studies frames and Bessel sequences in Fréchet spaces and their duals, connecting them to Schauder frames and representing systems, with practical applications.

Findings

Established definitions of frames and Bessel sequences in Fréchet spaces and duals.

Analyzed the relation between these frames and Schauder frames.

Provided applications to sampling sets and Dirichlet series in analytic function spaces.

Abstract

Frames and Bessel sequences in Fr\'echet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions.