Symmetric approximation of frames and bases in Hilbert spaces (Q2759077)

Let \(H\) be a Hilbert space and \(\{f_i\}_{i=1}^n\) a family of vectors in \(H\). Then \(\{f_i\}_{i=1}^n\) is a frame for its linear span, i.e., there exist constants \(A,B>0\) such that NEWLINE\[NEWLINE A\|f\|^2 \leq \sum_{i=1}^n |\langle f, f_i\rangle|^2 \leq B \|f\|^2NEWLINE\]NEWLINE for all \(f\in \text{span}\{f_i\}_{i=1}^n\). Defining \(Sf= \displaystyle\sum_{i=1}^n \langle f, f_i\rangle f_i\), the operator \(S\) is invertible on \(\text{span}\{f_i\}_{i=1}^n\), and NEWLINE\[NEWLINE f= \sum_{i=1}^n \langle f, S^{-1}f_i\rangle f_i, \quad f\in \text{span}\{f_i\}_{i=1}^n.NEWLINE\]NEWLINE If the frame is tight and normalized, i.e., if we can choose \(A=B=1\), then \(S=I\), and the cumbersome inversion of \(S\) needed to apply this formula in the general case is avoided. For this reason it is desirable to approximate given frames by normalized tight frames. A normalized tight frame \(\{\nu_i\}_{i=1}^n\) is called a symmetric approximation of the given frame \(\{f_i\}_{i=1}^n\) if \(\{\nu_i\}_{i=1}^n\) spans the same space as \(\{f_i\}_{i=1}^n\) and furthermore minimizes \(\sum_{i=1}^n \|f_i - \mu_i \|^2\) among all normalized tight frames \(\{\mu_i\}_{i=1}^n\) with this property. It is proved that a symmetric approximation exists and is unique. Furthermore, if \(\{f_i\}_{i=1}^n\) is linearly independent, then \(\{\nu_i\}_{i=1}^n\) is an orthonormal system. Extensions to frames \(\{f_i\}_{i=1}^\infty\) for infinite-dimensional subspaces of \(H\) are given.