The "test space and pairing" idea for frames and some generalized characterizations and topological properties of Euclidean continuous frames

Abstract: We introduce the "test space and pairing" idea for frames and apply it to the $\ell^2$ and $L^2$ spaces. First, we show that for every $J \neq \emptyset$, the notions of a classical $I$-frame with values in $H$ and a $J$-extended classical $I$-frame with values in $H$ are the same. The definition of a $J$-extended classical $I$-frame with values in $H$, $u = (u_i){i \in I}$, utilizes the "test space and pairing" idea by replacing the usual "test space" $H$ with $\ell^2(J;H)$ and the usual "pairing" ${P : (v ; (u_i){i \in I}) \in H \times \mathcal{F}I^H \mapsto (\langle v , u_i \rangle){i \in I} \in \ell^2(I;\mathbb{F})}$ with ${P^J : ((v_j){j \in J} ; (u_i){i \in I}) \in \ell^2(J;H) \times \mathcal{F}I^H \mapsto (\langle v_j , u_i \rangle){(i,j) \in I \times J} \in \ell^2(I \times J;\mathbb{F})}$. Secondly, we prove a similar result when the space $\ell^2(J;H)$ is replaced with the space $L^2(Y,\nu;H)$ and the frame $u = (u_x){x \in X}$ is $(X,\mu)$-continuous. Besides, we define the $J$-extended and $(Y,\nu)$-extended analysis, synthesis, and frame operators of the frame $u$ and note that they are just natural block-diagonal operators. After that, we generalize quite straightforwardly the well-known characterizations of Euclidean finite frames to the corresponding characterizations of Euclidean continuous frames. One tool that we use in this endeavor is some rewritings of the quotients $N(v ; (u_x){x \in X})$ and $N( (v_y){y \in Y}) ; (u_x){x \in X} )$. Besides, we give a simple sufficient condition for having a frame with values in $\mathbb{F}^2$ and use it to provide an example of a classical $\mathbb{N}^*$-frame with values in $\mathbb{C}^2$. Finally, we generalize some topological properties of the set of frames and Parseval frames from the Euclidean finitely indexed case to the Euclidean continuous one.