Frame approximation of pseudo-inverse operators (Q5946928)

A family of elements \(\{f_i\}_{i\in I}\) in a Hilbert space \({\mathcal H}\) is said to be a frame for \({\mathcal H}\) if there are constants \(A,B> 0\) such that NEWLINE\[NEWLINEA\|f\|^2\leq \sum_{i\in I}|\langle f,f_i\rangle|^2\leq B\|f\|^2NEWLINE\]NEWLINE for every \(f\in{\mathcal H}\). (Through the paper \({\mathcal H}\) is assumed to be separable.)NEWLINENEWLINENEWLINELet \(T\) be an operator acting on \(({\mathcal H},\langle\cdot,\cdot\rangle)\) and let \(\{f_i\}^\infty_{i\in 1}\) be a frame for the orthogonal complement of the kernel of \(T\). A sequence of operators \(\{\Phi_n\}\), \(\Phi_n(\cdot)= \sum^n_{i=1} \langle\cdot, g^n_i\rangle f_i\) is constructed which converges to the pseudo-inverse \(T^+\) of \(T\) in the strong topology as \(n\to\infty\). The operators \(\{\Phi_n\}\) are found by finite-dimensional methods. Also an adaptive iterative version of that result is given.