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http://dx.doi.org/10.4134/JKMS.2011.48.5.899

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

Gol, Rajab Ali Kamyabi (Department of Pure Mathematics Ferdowsi University of Mashhad Center of Excellence in Analysis on Algebraic Structures (CEAAS))
Tousi, Reihaneh Raisi (Department of Pure Mathematics Ferdowsi University of Mashhad)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.5, 2011 , pp. 899-912 More about this Journal
Abstract
We introduce ${\varphi}$-frames in $L^2$(G), as a generalization of a-frames defined in [8], where G is a locally compact Abelian group and ${\varphi}$ is a topological automorphism on G. We give a characterization of ${\varphi}$-frames with regard to usual frames in $L^2$(G) and show that ${\varphi}$-frames share several useful properties with frames. We define the associated ${\varphi}$-analysis and ${\varphi}$-preframe operators, with which we obtain criteria for a sequence to be a ${\varphi}$-frame or a ${\varphi}$-Bessel sequence. We also define ${\varphi}$-Riesz bases in $L^2$(G) and establish equivalent conditions for a sequence in $L^2$(G) to be a ${\varphi}$-Riesz basis.
Keywords
${\varphi}$-bracket product; ${\varphi}$-factorable operator; ${\varphi}$-frame; ${\varphi}$-Riesz basis; locally compact Abelian group;
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