Approximation of the inverse frame operator and applications to Gabor frames (Q1976278)

Recall that a countable family \(f_i\) in a Hilbert space is a frame if \(A\|f\|^2\leq\sum|\langle f,f_i\rangle|^2\leq B\|f\|^2\). The frame operator is \(Sf=\sum\langle f,f_i\rangle f_i\). It is known that \(f=\sum\langle f,S^{-1}f_i\rangle f_i\). If \(H_n=\text{span}\{f_1,\ldots,f_n\}\) and \(S_n=\sum_{i\leq n}\langle f,f_i\rangle f_i\), it is proved that for a large class of frames (containing Gabor frames) \(\langle f,S_n^{-1}f_i \rangle\not\to\langle f,S^{-1}f_i \rangle\). The authors show that, however, there exists a sequence \(m(n)\), depending only on \(f_i\) (and which can be calculated by finite-dimensional systems) such that for the orthogonal projection \(P_n\) onto \(H_n\) the relation \((P_nS_{n+m(n)})^{-1}P_nf\to S^{-1}f\) holds. Note that the calculation of the approximating operators is essentially a finite-dimensional problem. For the frame coefficients, one has convergence \(\langle f,(P_nS_{n+m(n)})^{-1}P_nf_i \rangle\to \langle f,S^{-1}f_i\rangle\) even in \(\ell_2\). Also for the best approximating solution of the moment problem one has a corresponding convergence. For all convergence results explicit error estimates are given. Also a different sequence \(m(n)\) is studied which for a similar kind of approximations gives better error estimates. The results are applied to Gabor frames.