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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)
  • Received : 2009.05.18
  • Accepted : 2009.10.08
  • Published : 2011.09.01

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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