Generalized shift-invariant systems and approximately dual frames
Dual pairs of frames yield a procedure for obtaining a perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force to consider nonperfect reconstructions. It is then important to study the deviation from the perfect reconstruction and to give tools to estimate it. The present paper deals especially with the case of generalized shift-invariant systems \(f_k (k\in\mathbb{N})\), \(g_k (k\in\mathbb{N})\) in \(L^2(\mathbb{R})\) and the estimation of the deviation is expressed by use of the operator norm \[ \left\| I-\sum_{k=1}^{+\infty}<.,g_k>f_k\right\|\;=\;\sup_{\|f\|=1}\left\|f-\sum_{k=1}^{+\infty}<f,g_k>f_k\right\|. \] It generalizes some known results and provides new ones. The paper is clearly written and structured. The list of references gives a very good idea of the current research in the domain and shows how the results of the paper can be integrated.
scientific article; zbMATH DE number 6694406