Series expansions in Fréchet spaces and their duals; construction of Fréchet frames

Abstract

Frames for Fréchet spaces $X_F$ with respect to Fréchet sequence spaces ${\Theta }_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^{\ast }$, are determined. If $\{g_i\}$ is a ${\Theta }_0$-frame for $X_0$, we construct a sequence $\{X_s{\}}_{s\in {\mathbb{N}}_0}$, $X_s\subset X_{s-1}$, $s\in \mathbb{N}$, for given ${\Theta }_F$, respectively a sequence $\{{\Theta }_s{\}}_{s\in {\mathbb{N}}_0}$, ${\Theta }_s\subset {\Theta }_{s-1}$, $s\in \mathbb{N}$, for given $X_F$, so that $\{g_i\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to ${\Theta }_F$ under different assumptions given on $X_0$, ${\Theta }_0$, ${\Theta }_F$ or $X_0$, ${\Theta }_0$, $X_F$.

Cited by in corpus (5)

• Characterization of atomic decompositions, Banach frames, Xd-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theory
• On sequence spaces for Fréchet frames
• Shrinking and boundedly complete atomic decompositions in Fréchet spaces
• Frames, semi-frames, and Hilbert scales
• Invariance of Fréchet Frames under Perturbation