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Equal-norm tight frames with erasures. (English) Zbl 1035.42029
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 $f\in H$: $$ A \Vert f\Vert ^2 \le \sum_{i \in I} \vert \langle f, e_i \rangle \vert ^2 \le B \Vert f\Vert ^2.$$ 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.

42C40Wavelets and other special systems
42C15General harmonic expansions, frames
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