φ-FRAMES AND φ-RIESZ BASES ON LOCALLY COMPACT ABELIAN GROUPS

Title & Authors
φ-FRAMES AND φ-RIESZ BASES ON LOCALLY COMPACT ABELIAN GROUPS
Gol, Rajab Ali Kamyabi; Tousi, Reihaneh Raisi;

Abstract
We introduce $\small{{\varphi}}$-frames in $\small{L^2}$(G), as a generalization of a-frames defined in [8], where G is a locally compact Abelian group and $\small{{\varphi}}$ is a topological automorphism on G. We give a characterization of $\small{{\varphi}}$-frames with regard to usual frames in $\small{L^2}$(G) and show that $\small{{\varphi}}$-frames share several useful properties with frames. We define the associated $\small{{\varphi}}$-analysis and $\small{{\varphi}}$-preframe operators, with which we obtain criteria for a sequence to be a $\small{{\varphi}}$-frame or a $\small{{\varphi}}$-Bessel sequence. We also define $\small{{\varphi}}$-Riesz bases in $\small{L^2}$(G) and establish equivalent conditions for a sequence in $\small{L^2}$(G) to be a $\small{{\varphi}}$-Riesz basis.
Keywords
$\small{{\varphi}}$-bracket product;$\small{{\varphi}}$-factorable operator;$\small{{\varphi}}$-frame;$\small{{\varphi}}$-Riesz basis;locally compact Abelian group;
Language
English
Cited by
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