Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Given a Hilbert space \(H\), a sequence \(\{ e_j : j \in J\} \subset H\) is a frame if there exist constants \(A, B >0\) such that for each \(f\in H\): \[ A \| f \| ^2 \leq \sum _{j \in J} | \langle f, e_j \rangle | ^2\leq B \| f \| ^2. \] A family \(\{ \phi_j : j \in J\} \subset H\) is a Riesz frame if it is complete and if there are constants \(A, B>0\) such that for any subset \(I \subset J\), \(\{ \phi_i : i \in I\}\) is a frame for the closure of its span with frame bounds \(A\) and \(B\). \(\{ \phi_j : j \in J\}\) is a Riesz sequence if there exist constants \(A, B>0\) such that for all subsets \(I \subset J\) and all square summable sequences \(\{c_i:i \in I\}\), \[ A \sum_{i \in I} | c_i| ^2 \leq \left\| \sum_{i \in I} c_i \phi_i \right\| ^2 \leq B \sum_{i \in I} | c_i| ^2. \] The authors prove that every Riesz frame can be partitioned into a finite union of Riesz sequences, and that the number of these sequences can be controlled by the ratio of the frame bounds. Various characterizations of Riesz frames are provided and the characterizing terms include decompositions into Riesz sequences, linearly independent sets, and conditions on the angles between the frame elements. It is also shown that compactly supported wavelets can be decomposed into a finite union of linearly independent sets.
scientific article; zbMATH DE number 1864738