Optimal dual frames for erasures
Let \(X=\{x_1,\dots,x_n\}\) be a frame for a \(k\)-dimensional Hilbert space \(H\), where \(n>k\). In coding theory, a signal \(x\in H\) is encoded as \(\Theta(x) = (\langle x, x_j \rangle)_{j=1}^n\). Then \(\Theta(x)\) is sent to a receiver for decoding to reconstruct \(x\). But some coefficients of \(\Theta(x)\) may be lost in the transmission process. The authors reconstruct \(x\) using a dual frame that minimizes the error of reconstruction. Such a dual frame is called \textit{optimal dual frame for erasures}. It is shown that in general the canonical dual frame of \(X\) is not an optimal dual frame for erasures. Further, optimal dual frames are not necessarily unique. The authors present sufficient conditions (e.g., \(X\) is a tight uniform frame or a group frame) under which the canonical dual frame of \(X\) is the unique optimal dual frame for erasures.
scientific article; zbMATH DE number 5650054