Fréchet frames, general definition and expansions

Abstract

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, || ̇ ||1 ≤ || ̇ ||2, and a BK-space (Θ, ||| ̇ |||). Then by the use of decreasing sequences of Banach spaces {Xs}s=0∞ and of sequence spaces {Θs} ∞s=0, we define a General Fréchet frame on the Fréchet space XF = ∩s=0∞ Xs. We obtain frame expansions of elements of XF and its dual X*F, as well of some of the generating spaces of X F with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations. © 2014 World Scientific Publishing Company. © 2014 World Scientific Publishing Company.