The equivalence of \(F_{a}\) -frames

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

[学科分类] 电力

[关键词] Frame;\(F_{a}\) -Frame;\(F_{a}\) -Equivalence