Fréchet frames, general definition and expansions

Abstract

We define an {\it $(X_1,\Theta ,X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot {|}_1\le |\cdot {|}_2$, 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 ${{\Theta }_s}_{s=0}^{\infty }$, 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 $X_F^{\ast }$, 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 $U:X_F\to {\Theta }_F$.

A new section is added and a minor revision is done

References in corpus (1)

• On sequence spaces for Fréchet frames

Cited by in corpus (2)

• Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing
• Frame expansions of test functions, tempered distributions, and ultradistributions