Finite-dimensional approximation of the inverse frame operator (Q1976489)

A frame in a Hilbert space \(H\) allows every element of \(H\) to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of these coefficients and many other situations where frames occur, require knowledge of the inverse frame operator, but it is usually difficult to invert the frame operator if the underlying Hilbert space is infinite--dimensional. In this paper the author introduces a method of approximation of the frame operator using finite subsets of the frame. In particular, this allows one to approximate the frame coefficients (even in the \(\ell^2\) sense) using finite--dimensional linear algebra. The author also shows that the method can be simplified in the important cases of Weil--Heisenberg frames and wavelet frames.