Journal of the
Korean Mathematical Society JKMS

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

MSC numbers: Primary 43A15; Secondary 43A25, 42C15