The equivalence of $F_{a}$-frames

Abstract

AbstractStructured frames such as wavelet and Gabor frames in $L^{2}(\mathbb {R})$L2(R) have been extensively studied. But $L^{2}(\mathbb{ R}_{+})$L2(R+) cannot admit wavelet and Gabor systems due to $\mathbb{R}_{+}$R+ being not a group under addition. In practice, $L^{2}(\mathbb{R}_{+})$L2(R+) models the causal signal space. The function-valued inner product-based $F_{a}$Fa-frame for $L^{2}(\mathbb{R}_{+})$L2(R+) was first introduced by Hasankhani Fard and Dehghan, where an $F_{a}$Fa-frame was called a function-valued frame. In this paper, we introduce the notions of $F_{a}$Fa-equivalence and unitary $F_{a}$Fa-equivalence between $F_{a}$Fa-frames, and present a characterization of the $F_{a}$Fa-equivalence and unitary $F_{a}$Fa-equivalence. This characterization looks like that of equivalence and unitary equivalence between frames, but the proof is nontrivial due to the particularity of $F_{a}$Fa-frames.