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arXiv · 1201.2096

Fréchet frames, general definition and expansions

Stevan Pilipović·Diana T. Stoeva
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Abstract

We define an {\it $(X_1,Θ, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space $(Θ, \snorm[\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${Θ_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' echet frame to be a general Fr\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\toΘ_F$.

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Stevan Pilipović, Diana T. Stoeva. 2012-08-09. Fréchet frames, general definition and expansions. https://doi.org/10.1142/s0219530514500018

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