A note on the description of frames of general form
A frame for a separable Hilbert space \(H\) is a collection \(\{f_i: i \in N\} \subset H\) such that there exist constants \(0 < A \leq B < \infty\) so that for all \(g \in H\), \[ A\| g\| _H^2 \leq \sum_{i \in N}| \langle g, f_i \rangle| ^2 \leq B\| g\| _H^2. \] The constants \(A, B\) are called the frame bounds. When \(A=B\) we say that the frame is tight. A theorem due to Naimark states that if \(\{f_i: i \in N\}\) is a tight frame for a Hilbert space \(H\), then there exists a Hilbert space \(H'\), \(H\subset H'\), and an orthonormal basis \(\{e_i: i \in N\}\) for \(H'\) such that \(\{f_i: i \in N\}\) is an orthogonal projection of \(\{e_i: i \in N\}\) onto \(H\). The paper under review extends this result to hold for general frames (with \(A \leq B\)) and for Riesz bases, i.e., images of orthonormal bases under bounded and invertible transformations. For related results see, e.g., \textit{P.G. Casazza, D. Han} and \textit{D. R. Larson} [Contemp. Math. 247, 149--182 (1999; Zbl 0947.46010)].
scientific article; zbMATH DE number 1928353