Approximation of the frame coefficients using finite dimensional methods

A frame is a family {fi}i=1 of elements in a Hilbert space H with the property that every element in H can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on H. The projection method is introduced in order to avoid this problem. The basic idea is to consider finite subfamilies {fi}i=1 of the frame and the orthogonal projection Pn onto span{fi}i=1. For f ∈ H, Pnf has a representation as a linear combination of fi, i = 1, 2, ..n, and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n → ∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment problem. It turns out, that the class of “well-behaving frames” are identical for the two problems we consider.