The equivalence of F a $F_{a}$ -frames

Abstract

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

Keywords

• Frame
• F a $F_{a}$ -Frame
• F a $F_{a}$ -Equivalence