On the duality principle by Casazza, Kutyniok, and Lammers (Q636818)

In this interesting paper, the authors study duality relations in the frame theory. Let \(\{f_i\}_{i\in I}\) be a frame for a separable Hilbert space \(H\), where \(I\) denotes a countable index set. Then the Riesz-dual sequence \(\{\omega_j\}_{j\in I}\) of \(\{f_i\}_{i\in I}\) is defined by \[ \omega_j = \sum_{i \in I} \langle f_i, e_j\rangle h_i \qquad (j \in I), \] where \(\{e_j\}_{j\in I}\) and \(\{h_i\}_{i\in I}\) are orthonormal bases of \(H\). The notion of a Riesz-dual sequence was introduced by \textit{P. G. Casazza}, \textit{G. Kutyniok} and \textit{M. C. Lammers} [J. Fourier Anal. Appl. 10, No. 4, 383--408 (2004; Zbl 1058.42020)]. In the paper under review, the authors solve the following problem: Let \(\{f_i\}_{i\in I}\) be a frame for \(H\) and \(\{\omega_j\}_{j\in I}\) a Riesz sequence in \(H\). Under what conditions do there exist orthonormal bases \(\{e_j\}_{j\in I}\) and \(\{h_i\}_{i\in I}\) for \(H\) such that \(\{\omega_j\}_{j\in I}\) is a Riesz-dual sequence of \(\{f_i\}_{i\in I}\)? It is shown that the Riesz-dual sequences \(\{\omega_j\}_{j\in I}\) can be characterized in terms of frame properties of an associated sequence \(\{n_i\}_{i\in I}\). Several examples illustrate the results. Finally, the authors improve the Cauchy-Schwarz inequality: If \(\{f_i\}_{i\in I}\) is a Bessel sequence with bound \(B\), then \[ |\langle f_i,f_j \rangle |^2 \leq B\,(B - \|f_i\|^2 - \|f_j\|^2) + \|f_i\|^2\, \|f_j\|^2. \]