Equal-norm tight frames with erasures
Given a Hilbert space \(H\), a collection \(\{e_i: i \in I \} \subset H\) is called a frame if there exist constants \(A, B>0\) such that for all \(f\in H\): \[ A \| f\| ^2 \leq \sum_{i \in I} | \langle f, e_i \rangle | ^2 \leq B \| f\| ^2. \] When \(A=B\) we say that the frame is tight. When all elements \(e_i\) have the same norm, we say the frame is equal-norm, and when this norm is 1 we say the frame is normalized. The paper under review studies equal-norm tight frames in spaces \(R^d\) and \(C^d\), equipped with the usual Euclidean inner product. Typical examples of such frames are harmonic frames, which consist of coordinates of the discrete Fourier transform. The authors show that all equal-norm tight frames generated by one or two unitary operators on \(R^d\) (resp., \(C^d\)) are generalized harmonic frames. The last section of the paper is devoted to the study of frames that remain frames after deletion (erasure) of a finite number of its elements. Such frames are called robust to erasures, and harmonic frames have this property. Characterization of frames robust to \(k\) erasures is given.
scientific article; zbMATH DE number 1896710